Introduction

A composite integer N whose digit sum S(N) is equal to the sum of the digits of its prime factors Sp (N) is called a Smith number [17].

For example 85 is a Smith number because digit sum of 85 i.e. S(85) = 8 + 5=13, which is equal to the sum of the digits of its prime factors i.e. Sp (85) = Sp (17 x 5) = 1 + 7 + 5 = 13.

Albert Wilansky named Smith numbers from his brother-in-law Harold Smiths telephone number 4937775 with this property i.e.

4937775 = 3.5.5.65837

Since

4+9+3+7+7+7+5=3+5+5+(6+5+8+3+7)=42

Wilansky also mentioned two other numbers with this property i.e. 9985 and 6036.Wilansky has found that there are 360 Smith numbers less than 10000, which is not correct, as there are 376 Smith numbers less then 10000. It is now known that there are infinitely many Smith numbers [2].

There are 25154060 smith numbers below 10⁹[4]. Further computations reveal that there are 241882509 smith numbers below 10¹⁰. Jens Kruse Andersen reported on 28th April 2008 that there are 2335807857 smith numbers below 10¹¹.


All 376 Smith numbers below 10000 are :

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086, 1111, 1165, 1219, 1255, 1282, 1284, 1376, 1449, 1507, 1581, 1626, 1633, 1642, 1678, 1736, 1755, 1776, 1795, 1822, 1842, 1858, 1872, 1881, 1894, 1903, 1908, 1921, 1935, 1952, 1962, 1966, 2038, 2067, 2079, 2155, 2173, 2182, 2218, 2227, 2265, 2286, 2326, 2362, 2366, 2373, 2409, 2434, 2461, 2475, 2484, 2515, 2556, 2576, 2578, 2583, 2605, 2614, 2679, 2688, 2722, 2745, 2751, 2785, 2839, 2888, 2902, 2911, 2934, 2944, 2958, 2964, 2965, 2970, 2974, 3046, 3091, 3138, 3168, 3174, 3226, 3246, 3258, 3294, 3345, 3366, 3390, 3442, 3505, 3564, 3595, 3615, 3622, 3649, 3663, 3690, 3694, 3802, 3852, 3864, 3865, 3930, 3946, 3973, 4054, 4126, 4162, 4173, 4185, 4189, 4191, 4198, 4209, 4279, 4306, 4369, 4414, 4428, 4464, 4472, 4557, 4592, 4594, 4702, 4743, 4765, 4788, 4794, 4832, 4855, 4880, 4918, 4954, 4959, 4960, 4974, 4981, 5062, 5071, 5088, 5098, 5172, 5242, 5248, 5253, 5269, 5298, 5305, 5386, 5388, 5397, 5422, 5458, 5485, 5526, 5539, 5602, 5638, 5642, 5674, 5772, 5818, 5854, 5874, 5915, 5926, 5935, 5936, 5946, 5998, 6036, 6054, 6084, 6096, 6115, 6171, 6178, 6187, 6188, 6252, 6259, 6295, 6315, 6344, 6385, 6439, 6457, 6502, 6531, 6567, 6583, 6585, 6603, 6684, 6693, 6702, 6718, 6760, 6816, 6835, 6855, 6880, 6934, 6981, 7026, 7051, 7062, 7068, 7078, 7089, 7119, 7136, 7186, 7195, 7227, 7249, 7287, 7339, 7402, 7438, 7447, 7465, 7503, 7627, 7674, 7683, 7695, 7712, 7726, 7762, 7764, 7782, 7784, 7809, 7824, 7834, 7915, 7952, 7978, 8005, 8014, 8023, 8073, 8077, 8095, 8149, 8154, 8158, 8185, 8196, 8253, 8257, 8277, 8307, 8347, 8372, 8412, 8421, 8466, 8518, 8545, 8568, 8628, 8653, 8680, 8736, 8754, 8766, 8790, 8792, 8851, 8864, 8874, 8883, 8901, 8914, 9015, 9031, 9036, 9094, 9166, 9184, 9193, 9229, 9274, 9276, 9285, 9294, 9296, 9301, 9330, 9346, 9355, 9382, 9386, 9387, 9396, 9414, 9427, 9483, 9522, 9535, 9571, 9598, 9633, 9634, 9639, 9648, 9657, 9684, 9708, 9717, 9735, 9742, 9760, 9778, 9840, 9843, 9849, 9861, 9880, 9895, 9924, 9942, 9968, 9975, 9985.

Note that the Beast number 666 is also a Smith Number.

   Various Kinds of Smith Numbers

  

• Smith Semiprimes:

  The smith numbers which are product of two prime numbers can be termed as Smith Semiprimes . For example 22 is a Smith Semiprime (Sloane's A098837).

  All Smith Semiprimes below 10⁴ are:

  4, 22, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 382, 391, 454, 517, 526, 535, 562, 634, 706, 778, 895, 913, 922, 958, 985, 1111, 1165, 1219, 1255, 1282, 1507, 1633, 1642, 1678, 1795, 1822, 1858, 1894, 1903, 1921, 1966, 2038, 2155, 2173, 2182, 2218, 2227, 2326, 2362, 2434, 2461, 2515, 2578, 2605, 2614, 2722, 2785, 2839, 2902, 2911, 2965, 2974, 3046, 3091, 3226, 3442, 3505, 3595, 3622, 3649, 3694, 3802, 3865, 3946, 3973, 4054, 4126, 4162, 4189, 4198, 4279, 4306, 4369, 4414, 4594, 4702, 4765, 4855, 4918, 4954, 4981, 5062, 5071, 5098, 5242, 5269, 5305, 5386, 5422, 5458, 5485, 5539, 5602, 5638, 5674, 5818, 5854, 5926, 5935, 5998, 6115, 6178, 6187, 6259, 6295, 6385, 6439, 6457, 6502, 6583, 6718, 6835, 6934, 7051, 7078, 7186, 7195, 7249, 7339, 7402, 7438, 7447, 7465, 7627, 7726, 7762, 7834, 7915, 7978, 8005, 8014, 8023, 8077, 8095, 8149, 8158, 8185, 8257, 8347, 8518, 8545, 8653, 8851, 8914, 9031, 9094, 9166, 9193, 9229, 9274, 9301, 9346, 9355, 9382, 9427, 9535, 9571, 9598, 9634, 9742, 9778, 9895, 9985 etc :

  

• Palindromic Smith Numbers:

  Smith numbers, which are also palindromic ( i.e. reading the same forward as well as backward) can be termed as Palindromic Smith numbers (Sloane's A098834).

  All Palindromic Smith numbers below 10⁶ are:

  4, 22, 121, 202, 454, 535, 636, 666, 1111, 1881, 3663, 7227, 7447, 9229, 10201, 17271, 22522, 24142, 28182, 33633, 38283, 45054, 45454, 46664, 47074, 50305, 51115, 51315, 54645, 55055, 55955, 72627, 81418, 82628, 83038, 83938, 90409, 95359, 96169, 164461, 173371, 239932, 256652, 262262, 294492, 362263, 373373, 445544, 454454, 505505, 515515, 535535, 545545, 635536, 704407, 717717, 832238, 841148, 864468, 951159, 956659, 974479 and 983389.

  

• Reversible Smith Numbers:

  Smith numbers, whose reversal is also a smith number, can be termed as Reversible Smith numbers (Sloane's A104171).

  All Reversible Smith numbers below 10⁴ are:

  4, 22, 58, 85, 121, 202, 265, 319, 454, 535, 562, 636, 666, 913, 1111, 1507, 1642, 1881, 1894, 1903, 2461, 2583, 2605, 2614, 2839, 3091, 3663, 3852, 4162, 4198, 4369, 4594, 4765, 4788, 4794, 4954, 4974, 4981, 5062, 5386, 5458, 5539, 5674, 5818, 5926, 6295, 6439, 6835, 7051, 7227, 7249, 7438, 7447, 8158, 8185, 8347, 8518, 8545, 8874, 8914, 9229, 9346, 9355, 9382, 9427 and 9634.

  Note that all palindromic smith numbers mentioned above are special case of reversible smith numbers.

  

• Fibonacci Smith Numbers:

  The fibonacci numbers, which are also smith numbers can be termed as Fibonacci Smith Numbers. During computations of smith numbers, it was found that the smallest fibonacci smith number is 1346369.

  F₃₁ = 1346269 = 557 x 2417

  Since

  1+3+4+6+2+6+9 = 5+5+7+2+4+1+7

  On further computations two more fibonacci smith numbers have been found, these are:

  F₇₇ = 5527939700884757 = 13 x 89 x 988681 x 4832521

  F₂₃₁ = 844617150046923109759866426342507997914076076194
  = 2 x 13 x 89 x 421 x 19801 x 988681 x 4832521 x 9164259601748159235188401

  It may be interesting study to find more fibonacci smith numbers.

  

• Abundant and Deficient Smith Numbers:

  Numbers such that s(n), the sum of aliquot divisors of n, is greater than n are called Abundant numbers. If n is also a smith number then it can be termed as Abundant Smith Number . For example 438 is a smith number and s(438) = 1 + 2 + 3 + 6 + 73 + 146 + 219 = 450, which is greater than 438. So 438 is a Abundant Smith Number (Sloane's A098835).

  All Abundant Smith numbers below 10⁴ are:

  378, 438, 576, 588, 636, 648, 654, 666, 690, 728, 762, 852, 1086, 1284, 1376, 1626, 1736, 1776, 1842, 1872, 1908, 1952, 1962, 2286, 2484, 2556, 2576, 2688, 2934, 2944, 2958, 2964, 2970, 3138, 3168, 3174, 3246, 3258, 3294, 3366, 3390, 3564, 3690, 3852, 3864, 3930, 4428, 4464, 4472, 4592, 4788, 4794, 4880, 4960, 4974, 5088, 5172, 5248, 5298, 5388, 5526, 5772, 5874, 5936, 5946, 6036, 6054, 6084, 6096, 6188, 6252, 6344, 6684, 6702, 6760, 6816, 6880, 7026, 7062, 7068, 7674, 7764, 7782, 7784, 7824, 7952, 8154, 8196, 8372, 8412, 8466, 8568, 8628, 8680, 8736, 8754, 8766, 8790, 8792, 8874, 9036, 9184, 9276, 9294, 9296, 9330, 9396, 9414, 9522, 9648, 9684, 9708, 9760, 9840, 9880, 9924, 9942 and 9968.

  Numbers such that s(n), the sum of aliquot divisors of n, is less than n are called Deficient numbers. If n is also a smith number then it can be termed as Deficient Smith Number . For example 22 is a smith number and s(22) = 1 + 2 + 11 = 14, which is less than 22. So 22 is a Deficient Smith number (Sloane's A098836).

  All Deficient Smith Numbers below 10⁴ are:

  4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 382, 391, 454, 483, 517, 526, 535, 562, 627, 634, 645, 663, 706, 729, 778, 825, 861, 895, 913, 915, 922, 958, 985, 1111, 1165, 1219, 1255, 1282, 1449, 1507, 1581, 1633, 1642, 1678, 1755, 1795, 1822, 1858, 1881, 1894, 1903, 1921, 1935, 1966, 2038, 2067, 2079, 2155, 2173, 2182, 2218, 2227, 2265, 2326, 2362, 2366, 2373, 2409, 2434, 2461, 2475, 2515, 2578, 2583, 2605, 2614, 2679, 2722, 2745, 2751, 2785, 2839, 2888, 2902, 2911, 2965, 2974, 3046, 3091, 3226, 3345, 3442, 3505, 3595, 3615, 3622, 3649, 3663, 3694, 3802, 3865, 3946, 3973, 4054, 4126, 4162, 4173, 4185, 4189, 4191, 4198, 4209, 4279, 4306, 4369, 4414, 4557, 4594, 4702, 4743, 4765, 4832, 4855, 4918, 4954, 4959, 4981, 5062, 5071, 5098, 5242, 5253, 5269, 5305, 5386, 5397, 5422, 5458, 5485, 5539, 5602, 5638, 5642, 5674, 5818, 5854, 5915, 5926, 5935, 5998, 6115, 6171, 6178, 6187, 6259, 6295, 6315, 6385, 6439, 6457, 6502, 6531, 6567, 6583, 6585, 6603, 6693, 6718, 6835, 6855, 6934, 6981, 7051, 7078, 7089, 7119, 7136, 7186, 7195, 7227, 7249, 7287, 7339, 7402, 7438, 7447, 7465, 7503, 7627, 7683, 7695, 7712, 7726, 7762, 7809, 7834, 7915, 7978, 8005, 8014, 8023, 8073, 8077, 8095, 8149, 8158, 8185, 8253, 8257, 8277, 8307, 8347, 8421, 8518, 8545, 8653, 8851, 8864, 8883, 8901, 8914, 9015, 9031, 9094, 9166, 9193, 9229, 9274, 9285, 9301, 9346, 9355, 9382, 9386, 9387, 9427, 9483, 9535, 9571, 9598, 9633, 9634, 9639, 9657, 9717, 9735, 9742, 9778, 9843, 9849, 9861, 9895, 9975 and 9985.

  

• Smith Square Numbers:

  Smith numbers, which are perfect squares, can be termed as Smith Square Numbers (Sloane's A098839).

  All Smith Square Numbers below 10⁷ are:

  4, 121, 576, 729, 6084, 10201, 17424, 18496, 36481, 51529, 100489, 124609, 184041, 195364, 410881, 559504, 674041, 695556, 732736, 887364, 896809, 966289, 988036, 1038361, 1190281, 1238769, 1726596, 1852321, 2166784, 2975625, 3407716, 3613801, 3663396, 3849444, 3888784, 3892729, 4088484, 4309776, 4330561, 4809249, 4875264, 4888521, 5031049, 5225796, 5391684, 5438224, 5461569, 5527201, 5978025, 6517809, 6630625, 6635776, 6780816, 6864400, 6969600, 7059649, 7273809, 7717284, 7868025, 7946761, 8048569, 8567329, 8573184, 8608356, 9150625, 9455625, 9678321 and 9960336.

  

• Smith Cubic Numbers:

  Smith numbers, which are perfect cubes, can be termed as Smith Cubic Numbers (Sloane's A098838).

  All Smith Cubic Numbers below 10¹⁰ are:

  27, 729, 19683, 474552, 7077888, 7414875, 8489664, 62099136, 112678587, 236029032, 246491883, 257259456, 279726264, 345948408, 463684824, 567663552, 638277381, 721734273, 766060875, 988047936, 1177583616, 1412467848, 2131746903, 2493326016, 2714704875, 3023464536, 3215578176, 3294646272, 3951805941, 4443297984, 4843965888, 4895680392, 6158676537, 8266914648, 8340725952, 8792838144 and 8831234763.

  

• Smith Triangular Numbers:

  Smith numbers, which are also triangular numbers, can be termed as Smith Triangular Numbers (Sloane's A098840).

  All Smith Triangular Numbers below 10⁷ are:

  378, 666, 861, 2556, 5253, 7503, 10296, 16653, 27261, 28920, 29890, 32896, 46056, 72771, 84255, 85905, 92235, 94395, 120786, 132870, 141778, 157641, 215496, 328455, 345696, 385881, 386760, 396495, 424581, 529935, 533028, 588070, 654940, 682696, 683865, 723003, 778128, 866586, 885115, 897130, 941878, 959805, 977901, 993345, 1082656, 1134771, 1234806, 1398628, 1457778, 1466328, 1495585, 1528626, 1570878, 1597578, 1633528, 1680861, 1792671, 1875016, 1876953, 1925703, 1931595, 1983036, 2087946, 2089990, 2137278, 2273778, 2351196, 2396955, 2458653, 2692360, 2828631, 2859636, 2890810, 2924571, 2968266, 3296028, 3378700, 3383901, 3451878, 3467661, 3579150, 3654456, 3733278, 3757911, 3904615, 3935415, 4119885, 4444671, 4710915, 4887501, 4925091, 5492955, 5586153, 5592840, 5815755, 6725278, 6984453, 7051890, 7486515, 7505875, 7536903, 7583565, 7850703, 7858630, 8090253, 8158780, 8211378, 8280415, 8284485, 8341570, 8390656, 8398851, 8485140, 8501626, 8982441, 9281586, 9359301, 9541896, 9651421 and 9677800

  

• Repdigit Smith Numbers:

  Smith numbers, whose all digits are same ( i.e. repeated digits) can be termed as Repdigit Smith numbers (Sloane's A104166).

  All Repdigit Smith numbers below 10⁶⁰ are:

  4, 22, 666, 1111, 6666666, 4444444444, 44444444444444444444, 555555555555555555555555555, 55555555555555555555555555555555 and 4444444444444444444444444444444444444444444444444444444.

   Consecutive Smith Numbers

There are consecutive numbers which are all Smith numbers also, these can be termed as Consecutive Smith numbers. If there are 2 consecutive numbers which are Smith numbers, these are termed as Smith Brothers. The smallest set of 2 consecutive smith numbers i.e. Smith Brothers is (728,729).

All Smith Brothers below 10⁵ are:

(728, 729), (2964, 2965), (3864, 3865), (4959, 4960), (5935, 5936), (6187, 6188), (9386, 9387), (9633, 9634), (11695, 11696), (13764, 13765), (16536, 16537), (16591, 16592), (20784, 20785), (25428, 25429), (28808, 28809), (29623, 29624), (32696, 32697), (33632, 33633), (35805, 35806), (39585, 39586), (43736, 43737), (44733, 44734), (49027, 49028), (55344, 55345), (56336, 56337), (57663, 57664), (58305, 58306), (62634, 62635), (65912, 65913), (65974, 65975), (66650, 66651), (67067, 67068), (67728, 67729), (69279, 69280), (69835, 69836), (73615, 73616), (73616, 73617), (74168, 74169), (74298, 74299), (76495, 76496), (76911, 76912), (77385, 77386), (78744, 78745), (82488, 82489), (82640, 82641), (83744, 83745), (83928, 83929), (83937, 83938), (84759, 84760), (84882, 84883), (85135, 85136), (87362, 87363), (87855, 87856), (89743, 89744), (89904, 89905), (90228, 90229), (90872, 90873), (91255, 91256), (91364, 91365), (91488, 91489), (93275, 93276), (93471, 93472), (94094, 94095), (94184, 94185), (94584, 94585), (95277, 95278), (95984, 95985), (96151, 96152), (96921, 96922), (97915, 97916), (98022, 98023) and (98900, 98901) .

There are 615885 smith brothers below 10⁹.

If there are 3 consecutive numbers which are Smith numbers, these can be termed as Smith Triples. The smallest set of 3 consecutive smith numbers i.e. Smith Triples is (73615,73616,73617).

All Smith triples below 10⁶ are:

(73615, 73616, 73617), (209065, 209066, 209067), (225951, 225952, 225953), (283745, 283746, 283747), (305455, 305456, 305457), (342879, 342880, 342881), (656743, 656744, 656745), (683670, 683671, 683672), (729066, 729067, 729068), (747948, 747949, 747950), (774858, 774859, 774860), (879221, 879222, 879223) and (954590, 954591, 954592).

There are 15955 smith triples below 10⁹.

If there are 4 consecutive numbers which are Smith numbers, these can be termed as Smith Quads (or 4-consecutive Smith numbers). The smallest set of 4-consecutive smith numbers i.e. Smith Quads is (4463535, 4463536, 4463537, 4463538).

Smallest members of set of Smith Quads below 10⁸ are:

4463535, 6356910, 8188933, 9425550, 11148564, 15966114, 18542654, 21673542, 22821992, 23767287, 28605144, 36615667, 39227466, 47096634, 47395362, 48072396, 54054264, 55464835, 57484614, 57756450, 57761165, 58418508, 61843387, 62577157, 64572186, 65484066, 66878432, 67118680, 71845857, 75457380, 77247606, 78432168, 88099213, 89653781, 90166567 and 92656434.

There are 384 smith quads below 10⁹.

If there are 5 consecutive numbers which are Smith numbers, these can be termed as Smith Quints (or 5-consecutive Smith numbers). The smallest set of 5-consecutive smith numbers i.e. Smith Quints is (15966114, 15966115, 15966116, 15966117, 15966118).

Smallest members of set of Smith Quints below 10⁹ are:

15966114, 75457380, 162449165, 296049306, 296861735, 334792990, 429619207, 581097690, 581519244, 582548088, 683474015, 809079150, 971285861 and 977218716 .

There are 14 smith quints below 10⁹.

If there are k consecutive numbers which are Smith numbers, these can be termed as k-consecutive Smith numbers. The smallest set of 6-consecutive smith numbers is (2050918644, 2050918645, 2050918646, 2050918647, 2050918648, 2050918649). There is no set of higher consecutive smith numbers below 10¹⁰.

Can you find the smallest set of 7-consecutive smith numbers ?.

Jens Kruse Andersen reported on 28th April 2008 that

The smallest set of 7-consecutive smith numbers starts at 164736913905.

Below that there are 31 sets of 6 consecutive Smith numbers, starting at:

2050918644, 6826932280, 16095667238, 16214788810, 17753840815, 19627891048, 31894287635, 37417358132, 38327645947, 72635842286, 75725224588, 77924458232, 79735902902, 80490527739, 84911527648, 93497450408, 115397414704, 118266684888, 122256909967, 124374538831, 128551622624, 129440489539, 132638590595, 135169942385, 140820590944, 143570578744, 149563926065, 153903366948, 154627494580, 154833907731, 159822348654.

Can you find the smallest set of 8-consecutive smith numbers ?.

   k-Smith Numbers

In 1987, Wayne Mc Daniel proved that there are infinitely many Smith numbers[8]. Mc Daniel defined k-Smith numbers as the numbers for which the sum of the digits of the prime factors is equal to k multiplied by sum of the digits i.e. Sp(N)=k*S(N) and showed that there are infinitely many k-Smith numbers for each k by constructing an infinite sequence of them.

For example 42 is a 2-Smith number because digit sum of 42 i.e. S(42) = 4 + 2 = 6, and the sum of the digits of its prime factors i.e. Sp (42) = Sp (2*3*7) = 12. So, Sp(N)=2*S(N).(Sloane's A104390).

All 2-Smith Numbers below 10⁴ are:

32, 42, 60, 70, 104, 152, 231, 315, 316, 322, 330, 342, 361, 406, 430, 450, 540, 602, 610, 612, 632, 703, 722, 812, 1016, 1027, 1029, 1108, 1162, 1190, 1246, 1261, 1304, 1314, 1316, 1351, 1406, 1470, 1510, 1603, 2013, 2054, 2065, 2070, 2071, 2106, 2114, 2121, 2134, 2216, 2230, 2233, 2280, 2322, 2324, 2410, 2413, 2422, 2425, 2506, 2522, 2611, 2701, 2702, 3007, 3030, 3108, 3122, 3130, 3136, 3206, 3212, 3216, 3241, 3302, 3310, 3311, 3412, 3451, 3512, 3560, 3601, 3710, 4025, 4033, 4042, 4080, 4142, 4210, 4440, 4501, 4512, 5002, 5022, 5026, 5073, 5131, 5215, 5222, 5306, 5402, 5900, 6010, 6014, 6031, 6132, 6152, 6201, 6202, 6224, 6251, 6321, 7012, 7016, 7111, 7160, 7202, 7250, 7310, 7410, 7511, 8024, 8232, 8302, 9002, 9030, 9104 and 9322.

There are 2824664 2-Smith Numbers below 10⁹.

If there are composite numbers N such that Sp(N)=3*S(N). These can be termed as 3-Smith numbers (Sloane's A104391).

All 3-Smith Numbers below 10⁵ are:

402, 510, 700, 1113, 1131, 1311, 2006, 2022, 2130, 2211, 2240, 3102, 3111, 3204, 3210, 3220, 4031, 4300, 4410, 5310, 6004, 6100, 6300, 7031, 7120, 9000, 10034, 10125, 10206, 10251, 10304, 10413, 10521, 10612, 10800, 11033, 11111, 11114, 11116, 11121, 11141, 11305, 11520, 11800, 12150, 12240, 12304, 12311, 12320, 12502, 13005, 13022, 13031, 13034, 13223, 13301, 13302, 13320, 13410, 14032, 14141, 14320, 14500, 15022, 15030, 15100, 17101, 20016, 20031, 20052, 20114, 20153, 20213, 20216, 20301, 21030, 21052, 21111, 21250, 21251, 21510, 21511, 21520, 22015, 22202, 22230, 22300, 22410, 23001, 23002, 23022, 23130, 23140, 23212, 23500, 24011, 24021, 24100, 24500, 26011, 30005, 30015, 30051, 30131, 30221, 30303, 30304, 30331, 30413, 31061, 31211, 31230, 31300, 31521, 32004, 32201, 33022, 33100, 33320, 33400, 35100, 36001, 38000, 40040, 40132, 41013, 41400, 41500, 42003, 42100, 42120, 44003, 45030, 50020, 50031, 50032, 51101, 51113, 51120, 51121, 51220, 51400, 52003, 53010, 53400, 60100, 61012, 61201, 62410, 63101, 70021, 71010, 71020, 74000, 80122 and 90100

There are 147982 3-Smith Numbers below 10⁹.

If there are composite numbers N such that Sp(N)=4*S(N). These can be termed as 4-Smith numbers (Sloane's A103125).

All 4-Smith Numbers below 10⁵ are:

2401, 5010, 7000, 10005, 10311, 10410, 10411, 11060, 11102, 11203, 12103, 13002, 13021, 13101, 14001, 14101, 14210, 20022, 20121, 20203, 20401, 21103, 21112, 21120, 21201, 22040, 22101, 22201, 23030, 30003, 30031, 30320, 31002, 31101, 31111, 32101, 32200, 32300, 40002, 41101, 43000, 50001, 50211, 51200, 60001, 61000, 61110, 70002 and 71100 .

There are 13609 4-Smith Numbers below 10⁶.

If there are composite numbers N such that Sp(N)=5*S(N). These can be termed as 5-Smith numbers (Sloane's A103126).

All 5-Smith Numbers below 10⁶ are:

2030, 10203, 12110, 20210, 20310, 21004, 21010, 24000, 24010, 31010, 41001, 50010, 70000, 100004, 100012, 100210, 100310, 100320, 101020, 101041, 102022, 103200, 104010, 104101, 104110, 105020, 106001, 110020, 110202, 110212, 110400, 111013, 112002, 112030, 120010, 120050, 121030, 121201, 122001, 130200, 130210, 132000, 140011, 141010, 200030, 201103, 201111, 201131, 201202, 202003, 202030, 204010, 210210, 211030, 211101, 211120, 211220, 222010, 223010, 300004, 300013, 300300, 300310, 310400, 311001, 311011, 311110, 320010, 321010, 322000, 330020, 350000, 401010, 410002, 410011, 411010, 430000, 500002, 501011, 510110, 600010 and 610000.

There are 4204 5-Smith Numbers below 10⁹.

If there are composite numbers N such that Sp(N)=k*S(N). These can be termed as k-Smith numbers.

For K=6, 7 and 8 , the number of k-Smith numbers. below 10⁹ are 1238, 409 and 218 respectively.

For K=6, 7 and 8 , the smallest k-Smith numbers are 10112, 10 and 200 respectively.

   k^-1 -Smith Numbers

Mc Daniel [8] defined k^-1 -Smith number as composite integer N such that Sp(N)=k^-1*S(N) i.e. S(N)=k*Sp(N).

For example 88 is a 2^-1 -Smith number because digit sum of 88 i.e. S(88) = 8 + 8=16, which is equal to 2 times the sum of the digits of its prime factors i.e.2 x Sp (88) =2 x Sp (11 x 2 x 2 x 2) = 2 x( 1 + 1 + 2 + 2 + 2) = 16.(Sloane's A050224).

All 2^-1 -Smith Numbers below 10⁴ are:

88, 169, 286, 484, 598, 682, 808, 844, 897, 961, 1339, 1573, 1599, 1878, 1986, 2266, 2488, 2626, 2662, 2743, 2938, 3193, 3289, 3751, 3887, 4084, 4444, 4642, 4738, 4804, 4972, 4976, 4983, 5566, 5665, 5764, 5797, 5863, 5876, 6198, 6262, 6541, 6565, 6578, 6655, 6695, 6738, 6868, 6877, 6929, 7278, 7359, 7386, 7458, 7579, 7818, 7843, 7865, 7926, 7953, 7999, 8044, 8086, 8248, 8349, 8446, 8488, 8646, 8824, 8897, 8998, 9269, 9292, 9476, 9696, 9706, 9784, 9889, 9922, 9944 and 9988. .

There are 1087152 2^-1 -Smith Numbers below 10⁹.

If there are composite numbers N whose digit sum S(N) is equal to 3 times the sum of the digits of its prime factors i.e. S(N)=3*Sp(N). These can be termed as 3^-1 -Smith numbers (Sloane's A050225).

All 3^-1 -Smith Numbers below 10⁶ are:

6969, 19998, 36399, 39693, 66099, 69663, 69897, 89769, 99363, 99759, 109989, 118899, 181998, 191799, 199089, 297099, 306939, 333399, 336963, 339933, 363099, 396363, 397998, 399333, 399729, 588969, 606666, 606909, 639633, 660693, 666633, 678666, 693363, 693759, 696333, 759099, 759693, 786666, 787179, 798699, 828759, 834969, 897069, 915969, 928089, 938997, 975963, 990363 and 993729.

There are 6575 3^-1 -Smith Numbers below 10⁹.

If there are composite numbers N whose digit sum S(N) is equal to 4 times the sum of the digits of its prime factors i.e. S(N)=4*Sp(N). These can be termed as 4^-1 -Smith numbers (Sloane's A103123).

All 4^-1 -Smith Numbers below 10⁸ are:

19899699 , 36969999 , 36999699 , 39699969 , 39999399 , 39999993 , 66699699 , 66798798 , 67967799 , 67987986 , 69759897 , 69889389 , 69966699 , 69996993 , 76668999 , 79488798 , 79866798 , 85994799 , 86686886 , 89769759 , 89866568 , 93999993 , 98736968 and 99968199 .

There are 251 4^-1 -Smith Numbers below 10⁹.

If there are composite numbers N whose digit sum S(N) is equal to 5 times the sum of the digits of its prime factors i.e. S(N)=5*Sp(N). These can be termed as 5^-1 -Smith numbers (Sloane's A103124).

All 5^-1 -Smith Numbers below 10⁹ are:

399996663, 666609999, 669969663, 690696969 and 699966663 .

Mc Daniel in his paper[8] conjectured that there exist infinitely many k^-1 -Smith numbers for every k >1.

Can you prove the Mc Daniel Conjecture?.

   Construction of Smith Numbers

There is no general-purpose formula for generating all Smith numbers.

In 1983, Oltikar and Wayland [12] constructed Smith numbers using repunit primes. For example, 1540 · Rn is a Smith number for each repunit prime Rn. There are a lot of other possible multipliers, including

1540, 1720, 2170, 2440, 5590, 6040, 7930, 8344, 8470, 8920, 23590, 24490, 25228, 29080, 31528, 31780, 33544, 34390, 35380, 39970, 40870, 42490, 42598, 43480, 44380, 45955, 46270, 46810, 46990, 47908, 48790, 49960, 51490, 51625, 52345, 52570, 53290, 57070, 57160, 57880, 59770, 60625, 62146, 63928, 64360, 64540, 65080, 66970, 67528, 69355, 71380, 72982, 73810, 74440, 74710, 76780, 78040, 79030, 79570, 80470, 80740, 82270, 82990, 84790, 85330, 85870, 86590, 87490, 87580, 87850, 88228, 89560, 89740, 89830, 92440, 92620, 95266, 97030, 97048, 98875, 108955 ... (Sloane's A104167).

It has been found that all such multipliers have a digital root of 1 and also have a digital factor sum divisible by 9, for more details click here.

Multiplying any repunit prime Rn >11 by 3304 also produces smith numbers [6][12]. Oltikar and Wayland[12] mentioned that every prime whose digits are all 0 and 1 has some multiple that qualifies as a smith number [2].

In 1984, Pat Costello produced 75 Smith numbers of the form p · q · 10^k where p is a small prime and q is a Mersenne prime. The largest Smith number produced was

191 · (2²¹⁶⁰⁹¹ - 1) · 10²⁶⁶ consisting of 65319 digits.

In 1987, Wayne Mc Daniel made a big breakthrough when proving that there are infinitely many Smith numbers[8]. Mc Daniel introduced k-Smith numbers for which the sum of the digits of the prime factors is equal to k multiplied by sum of the digits and showed that there are infinitely many k-Smith numbers for each k by constructing an infinite sequence of them.

Kathy Lewis [2] produced another infinite sequence of Smith numbers of the form 11^k · 9 Rn · 10^m.

 

   Distribution of Smith Numbers v/s Primes

A computer program in Fortran has been developed to investigate Smith numbers. Objective of investigation was to know the proportion of Smith numbers as compared to prime numbers up to a given limit and maximum number of prime factors of Smith numbers. Let the number of Smith numbers up to x is denoted by SN(x) and number of primes up to x by pi(x). The values of SN(x)and pi(x) up to x = 10^m for m = 1,2,3, . . 11 are tabulated in Table 1. The ratio r = pi(x)/SN(x) is tabulated in column 4 of Table 1.

It is noted from Table 1 that percentage of Smith numbers decreases with increase in x. As an example, for x = 10⁵ , percentage of Smith numbers up to x = 3.294% which decreases to 2.9928% for x = 10⁶ . The percentage of primes also decreases with increase in x. It is observed that the ratio r decreases with increase in x for x = 10^m(for m>2), which indicate that the decrease in Smith numbers is at a slower rate, then decrease in prime numbers.

Table 1

x	SN(x)	pi(x)	pi(x)/ SN(x)
101	1	4	4.00000
102	6	25	4.16667
103	49	168	3.42857
104	376	1229	3.26862
105	3294	9592	2.91196
106	29928	78498	2.62289
107	278411	664579	2.38704
108	2632758	5761455	2.18837
109	25154060	50847534	2.02144
1010	241882509	455052511	1.88129
1011	2335807857	4118054813	1.76301

It is also noted during investigation that distribution of Smith numbers between 10^m and 10^m+1 generally presents a rising trend in this range. So generally SN(2 x 10^m ) - SN(10^m ) is lower than SN (10^m+1 ) - SN(9 x 10^m ).

As an example: for m = 7,

SN(2 x 10⁷ ) - SN (10⁷ ) = 527739 - 278411 = 249328

SN(10⁸ ) - SN(9 x 10⁷ ) = 2632758 - 2353482 = 279276

for m = 8,

SN(2 x 10⁸ ) - SN(10⁸ ) = 5029891 - 2632758 = 2397133

SN(10⁹ ) - SN (9 x 10⁸ ) = 25154060 - 22497704 = 2656356

This behavior of distribution of Smith numbers is different from prime numbers, which shows a falling trend. So it is likely that for large x, SN(x) may exceed pi(x) or in other words the ratio r may be less than 1. So we state the following Conjecture.

Conjecture: There exists a number x up to which, number of Smiths are equal to the number of Primes and also SN(10^m) > pi(10^m) for a value of m such that 10^m > x .

It may be an interesting study to find smallest value of x (which may be quite large, more than 10¹⁸) for which SN(x) = pi(x).

   Highly Decomposable Smith Numbers

Let the number of prime factors (counting multiplicity) of a number n is denoted by Ω(n), then n can be termed as Highly Decomposable number if for every n1 < n, Ω(n1) < Ω(n). It is obvious that the i^th Highly Decomposable number is 2ⁱ where, i=1,2,3,4...

Similarly , for every Smith number N1 < N, if Ω(N1) < Ω(N), then N can be termed as Highly Decomposable Smith number [4]. In other words, the Smith number which sets a record for number of prime factors, starting from the first Smith number can be termed as Highly Decomposable Smith number (Sloane's A104169). The Smallest Smith numbers with n prime factors (not necessarily distinct) are tabulated in Table 2(Sloane's A104168). It is found that the maximum number of prime factors of a Smith number up to 10¹⁰ is 27.

As can be seen from Table 2, the sequence of Highly Decomposable Smith numbers is

4, 27, 378, 576, 2688, 17496, 44928, 75776, 168960, 319488, 958464, 2883584, 5767168, 7077888, 279969792, 544997376, 778567680, 2579496960, 3875536896 . . .

It may be noted that except the number 27, all other Highly Decomposable Smith numbers are even.

Table 2

No. of Prime factors	Smallest Smith number
2	4
3	27
4	636
5	378
6	729
7	648
8	576
9	2688
10	17496
11	44928
12	75776
13	168960
14	765952
15	319488
16	958464
17	5537792
18	5963776
19	2883584
20	5767168
21	7077888
22	279969792
23	544997376
24	778567680
25	2579496960
26	4567597056
27	3875536896

Conjecture: All Highly Decomposable Smith numbers greater than 27 are even.

It may be interesting to find a counter example or to prove the above conjecture.

   Some Interesting Observations

• 31 is the smallest prime which is the sum of two smith numbers.
• Multiplying a number by 10 does not change the digit sum but add 7( = 2+5) to the sum of the digits of its prime factors.
• Smallest and largest Pandigital (i.e. containing all digits from 0 to 9) smith numbers are 1023465798 and 9876542103.
• Smallest and largest zeroless Pandigital (i.e. containing all digits from 1 to 9) smith numbers are 123456879 and 987653214.
• 728729 is a prime formed from concatenation of two consecutive smith numbers.
• 9R₁₀₃₁*(10²⁸⁵⁷²+8*10¹⁴²⁸⁶+1)¹⁰²⁷ * 10^2722434 is one of the largest known smith numbers consisting of 32066910 digits found by Patrick Costello[1].
• 9R₁₀₃₁ *(10⁶⁹⁸⁸²+3*10³⁴⁹⁴¹+1)¹⁴⁷⁶*10^3913210 is a smith number consisting of 107060074 digits.
• If m and n are positive integers , then Sp(mn) = Sp(m) + Sp(n) where Sp(n) denote the sum of digits of the prime factors of n. So, Sp is an additive function.
• A smith number with a digital root other than 4 must be the product of more than two primes.
• 2227 is the smallest smith number formed from concatenation of two consecutive smith numbers.
• 27, 58 and 85 are three consecutive Smith numbers and 27 + 58 = 85 .

References:

[1] Costello, Patrick. "A New Largest Smith Number," Fibonacci Quarterly, vol 40(4), 2002, pp. 369-371.

[2] Costello, Patrick and Lewis, Kathy. "Lots of Smiths," Mathematics Magazine, vol 75(3), 2002, pp. 223-226.

[3] Dudley,U. , "Smith Numbers", Mathematics Magazine, 67(1994),pp.62-65.

[4] Gupta, Shyam Sunder "Smith Numbers" Mathematical Spectrum ,37(2004/5), pp.27-29.

[5] Guy, R. K. "Smith Numbers." §B49 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 103-104, 1994.

[6] Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, pp. 205-206, 1998.

[7] McDaniel,W.L., "Palindromic Smith numbers", Journal of Recreational Mathematics, 19(1987), pp. 34-37.

[8] McDaniel,W.L., "The Existence of infinitely Many k- Smith numbers", Fibonacci Quarterly, 25(1987), pp.76-80.

[9] McDaniel,W.L., "Powerful k-Smith Numbers", Fibonacci Quarterly, 25(1987), pp.225-228.

[10] McDaniel,W.L. and Yates, Samuel, "The Sum of Digits Function and its Application to A Generalization of the Smith Number Problem", Nieuw Archief Voor Wiskunde, 7, No, 1-2, March/July, (1989), 39-51.

[11] McDaniel,W.L., "On the Intersection of the Sets of Base b Smith Numbers and Niven Numbers", Missouri J. of Math. Sci. 2 (1990), 132-136.

[12] Oltikar, Sham and Keith Wayland. "Construction of Smith Number," Mathematics Magazine, vol 56(1), 1983,
pp. 36-37.

[13] Pickover, Clifford A. "A Brief History of Smith Numbers." Ch. 104 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford University Press, pp. 247-248, 2001.

[14] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

[15] Sloane, N. J. A. Sequences A006753, A033662, A033663, A050218, A050224, A050225, A059754, A063844, A098834, A098835, A098836, A098837, A098838, A098839, A098840, A103123, A103124, A103125, A103126, A104166, A104167, A104168, A104169, A104170, A104171, A104390 and A104391, in "The On-Line Encyclopedia of Integer Sequences."

[16] Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1997.

[17] Wilansky, A. "Smith numbers",Two-Year College Mathematics Journal, 13(1982),p.21.

[18] Yates,S., "Smith numbers congruent to 4 (mod 9)", Journal of Recreational Mathematics, 19(1987), pp.139-141.

This page is created on 10 March, 2005.

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