Some Special Sphenic Numbers

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:$$27511=11 \times 41 \times 61$$This got me wondering how many sphenic numbers in the range up to 40,000 have this property. Well it turns out that 78 numbers do. These numbers are (permalink):

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:$$5842=2 \times 23 \times 127$$These numbers are (permalink):

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:$$5842=3 \times 13 \times 37$$These numbers are (permalink):

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:$$424883 = 41 \times 43 \times 241$$In the range up to 40,000, there is only one number that is sphenic and in which all three factors as well as the number itself share the digit 5 in common. This is the number:$$15635 = 5 \times 53 \times 59$$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 6 in common. If we consider the range up to one million, we find two numbers. The first of these is:$$666181 = 61 \times 67 \times 163$$There are 14 numbers that are sphenic and in which all three factors as well as the number itself share the digit 7 in common. The first such number is:$$7973 = 7 \times 17 \times 67$$These numbers are (permalink):

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:$$32509 = 19 \times 29 \times 59$$Before leaving, I'll return to the number that started all this: 27511. It has some other interesting properties involving prime numbers. These are:

• 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

The algorithm used earlier can be easily modified (permalink) to accommodate a number of distinct prime factors other than 3. In the case of four distinct prime factors, it is only the digit 3 that yields any numbers in the range up to 40,000. These numbers are:$$33189 = 3 \times 13 \times 23 \times 37 \\38571 = 3 \times 13 \times 23 \times 43$$Once the range is extended to one million, the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 have 132, 11, 277, 0, 0, 0, 22, 0 and 0 corresponding numbers respectively.