What struck me about the number associated with my diurnal age today was that it is sphenic and all three factors as well as the number itself share one digit in common, namely the digit 1. The number is:
1
2431, 2717, 4199, 6851, 9061, 10013, 10127, 10153, 11407, 12749, 13243, 13277, 13481, 13981, 14443, 14729, 14839, 15067, 15301, 15587, 15691, 16159, 16523, 17537, 18161, 18733, 18887, 19261, 19591, 19703, 19877, 20801, 21109, 21131, 21307, 21527, 21593, 21607, 22321, 22451, 22781, 23617, 23881, 24149, 24211, 25441, 25619, 27313, 27511, 27911, 28171, 28613, 28951, 29051, 30173, 30481, 30719, 31141, 31229, 31369, 31559, 32021, 32147, 32351, 32513, 32813, 33371, 34441, 34561, 35123, 35717, 36091, 36157, 37169, 37213, 37411, 37417, 39919
Naturally, I decided to investigate the remaining digits from 2 to 9. Here is what I found.
2
For the digit 2, there are 22 numbers that are sphenic and in which all three factors as well as the number itself share the digit 2 in common. The first such number is 5842:
5842, 10258, 10442, 11822, 12098, 12238, 12374, 12466, 12742, 12926, 12934, 13282, 13862, 15254, 15602, 16298, 23966, 24058, 24418, 30218, 33442, 38042
3
For the digit 3, there are 138 numbers that are sphenic and in which all three factors as well as the number itself share the digit 3 in common. The first such number is:
1443, 2139, 2553, 3237, 3441, 3657, 3999, 4773, 5037, 5343, 5883, 6357, 6837, 8103, 9039, 9213, 9321, 9453, 11037, 11063, 11433, 11937, 11973, 12183, 12363, 12543, 13143, 13197, 13287, 13317, 13533, 13611, 13767, 14313, 14937, 15387, 15483, 16377, 17329, 17673, 17931, 18093, 19203, 20397, 20683, 20739, 21183, 21359, 21423, 21783, 21873, 22317, 22839, 23127, 23253, 23907, 23943, 24357, 24753, 25323, 25493, 25737, 25863, 26319, 26381, 26637, 27393, 28137, 28923, 29193, 29739, 30003, 30057, 30147, 30291, 30441, 30567, 30659, 30687, 30783, 30797, 30831, 31341, 31413, 31947, 32097, 32271, 32457, 32523, 32619, 32721, 32829, 33267, 33387, 33449, 33657, 33787, 33927, 34077, 34113, 34131, 34437, 34521, 34611, 34689, 34707, 34743, 34917, 35113, 35187, 35247, 35457, 35619, 35697, 36087, 36177, 36507, 36543, 36593, 36741, 36921, 37047, 37167, 37407, 37789, 37797, 37887, 38001, 38337, 38517, 38739, 38847, 39169, 39183, 39507, 39603, 39849, 39923
4 to 9
There are no numbers in the range up to 40,000 that are sphenic and in which all three factors as well as the number itself share the digit 4 in common. If we consider the range up to one million, we find 20 numbers. The first of these is:
7973, 8687, 12173, 12733, 17353, 18907, 19873, 20587, 24017, 27013, 27713, 34237, 37051, 37723
For the digit 8, there are no sphenic numbers that satisfy even in the range up to one million. For the digit 9, there is only one number in the range up to 40,000 that satisfies and that is:
- number + sum of digits is prime: 27511 + 16 = 27527
- number + product of digits is prime: 27511 + 70 = 27581
- concatenation of prime factors in ascending order is prime: 114161
- concatenation 116141 is also prime
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