A Multiplicity of Digits: Part 2

A variation on the theme of my previous post, that also involves the multiple occurrence of the same digits, are these numbers that comprise a sequence that I've referenced as S107 in my Bespoken for Sequences database. 

Sphenic numbers containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first example of such a number is 1443 = 3 * 13 * 37 with a digital root of 3. There are 61 such numbers in the range up to 40000. Here is the list (permalink):

1443, 3441, 3657, 3999, 4773, 6357, 8103, 9039, 9453, 11037, 11433, 11937, 11973, 13143, 13197, 13287, 13611, 14313, 15483, 17931, 18093, 20397, 20739, 21423, 21783, 21873, 23907, 23943, 24357, 24753, 26319, 28137, 29739, 30441, 30567, 30783, 31341, 31413, 32097, 32457, 32619, 33267, 34077, 34113, 34131, 34437, 34689, 34707, 34743, 35247, 35697, 36507, 36543, 36741, 36921, 37047, 37407, 38001, 38739, 38847, 39603

A twist on this theme is to consider sphenic numbers that do NOT contain the digit 3. Such numbers could be considered as having a hidden multiplicity of digits because the prevalence of the digit is not immediately obvious. The same could be said of the numbers just mentioned but those cases the repeating digit is overtly visible. Here is the revised criteria:

Sphenic numbers NOT containing the digit 3 whose three prime factors also contain the digit 3 and whose additive digital root is 3.

The first such number is 1209 = 3 * 13 * 31 with a digital root of 3. There are 45 such numbers in the range up to 40000. Here they are (permalink):

1209, 1677, 2847, 4017, 5421, 5727, 6789, 7527, 7797, 8697, 9417, 9579, 12207, 12909, 12927, 14547, 14781, 15159, 15429, 16077, 16491, 16887, 17121, 17949, 17967, 18057, 18147, 20217, 20829, 21027, 21459, 22557, 24609, 24771, 24897, 25077, 26247, 26427, 26841, 27507, 28551, 28587, 28767, 28821, 29109

Rather than sphenic numbers, with three distinct prime factors, we could consider biprimes or numbers with two distinct prime factors. Firstly let's look at numbers with properities as follows:

Biprimes containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 73 such numbers in the range up to 40000. The first of these is 254 = 2 * 127 with a digital root of 2. Here they are (permalink):

254, 542, 2558, 2594, 2846, 3242, 4286, 4322, 4502, 4682, 5042, 5267, 5294, 5582, 5942, 8462, 12242, 12422, 12458, 12494, 12854, 14258, 16526, 17246, 17642, 18254, 18929, 19442, 20486, 21458, 22502, 22574, 23483, 23654, 24014, 24041, 24086, 24194, 24482, 24554, 24842, 24914, 25022, 25094, 25166, 25202, 25238, 25274, 25526, 25562, 25598, 25706, 25778, 25814, 25958, 25967, 26498, 26534, 27254, 27623, 28442, 28451, 28586, 28829, 29693, 29846, 30242, 30521, 32546, 33842, 35246, 36254, 39629

Again we can consider the revised criteria:

Biprimes NOT containing the digit 2 whose two prime factors also contain the digit 2 and whose additive digital root is 2. 

There are 53 such numbers in the range up to 40000 with the first being 1046 = 2 * 523 with a digital root of 2. Here are the numbers (permalink):

1046, 1658, 3683, 4034, 4106, 4178, 4358, 4538, 4574, 4754, 4853, 4934, 5006, 5078, 5114, 5186, 5366, 5438, 5834, 5906, 6509, 6518, 7058, 7454, 7859, 10046, 11054, 11846, 13646, 14438, 14474, 15167, 15446, 16418, 17858, 19658, 30854, 31646, 33401, 34058, 34418, 34598, 35687, 35858, 36434, 36506, 36578, 37046, 37091, 37613, 38414, 38846, 39854