A Multiplicity of Digits: Part 1

An interesting property of the number associated with my diurnal age yesterday was the following:$$27505=5 \times 5501$$The property is so obvious that you might miss it at first glance but I noticed it and created a new sequence based on it. It's recorded as S107 in my Bespoken for Sequences database. See Figure 1.

Figure 1

Let's restate the property that members of this sequence have:

Composite numbers containing the digit 5 at least once whose prime factors each contain the digit 5 as well so that, overall, the digit 5 occurs five times.

In the range up to 40000, there are 27 numbers that satisfy this criteria and they are (permalink):

5255, 5755, 7555, 12755, 15635, 17555, 25055, 25295, 25505, 25535, 25565, 25595, 25765, 25855, 26755, 27505, 27515, 27535, 27595, 27655, 27785, 27955, 28255, 28555, 29255, 32755, 35755

Most, with one exception, are biprimes with 5 as one of the two factors:

5255 = 5 * 1051 with total of 5

5755 = 5 * 1151 with total of 5

7555 = 5 * 1511 with total of 5

12755 = 5 * 2551 with total of 5

15635 = 5 * 53 * 59 with total of 5

17555 = 5 * 3511 with total of 5

25055 = 5 * 5011 with total of 5

25295 = 5 * 5059 with total of 5

25505 = 5 * 5101 with total of 5

25535 = 5 * 5107 with total of 5

25565 = 5 * 5113 with total of 5

25595 = 5 * 5119 with total of 5

25765 = 5 * 5153 with total of 5

25855 = 5 * 5171 with total of 5

26755 = 5 * 5351 with total of 5

27505 = 5 * 5501 with total of 5

27515 = 5 * 5503 with total of 5

27535 = 5 * 5507 with total of 5

27595 = 5 * 5519 with total of 5

27655 = 5 * 5531 with total of 5

27785 = 5 * 5557 with total of 5

27955 = 5 * 5591 with total of 5

28255 = 5 * 5651 with total of 5

28555 = 5 * 5711 with total of 5

29255 = 5 * 5851 with total of 5

32755 = 5 * 6551 with total of 5

35755 = 5 * 7151 with total of 5

I then thought about the other digits and a generalisation of the property, namely:

Composite numbers containing the digit \(d\) at least once whose prime factors  each contain the digit \(d\) as well so that, overall, the digit \(d\) occurs \(d\) times.

With the digit 1, it's pretty obvious that no numbers qualify so let's move on to the digit 2 where there are no numbers again that qualify (even in the range up to one million). With the digit 3, there are 448 numbers that qualify so I won't list them here. I'll just provide this permalink instead.

With the digit 4, there are three numbers that qualify: 16441, 18409 and 19049. Their factorisations are as follows (permalink):

16441 = 41 * 401 with total of 4

18409 = 41 * 449 with total of 4

19049 = 43 * 443 with total of 4

The digit 5 we have already covered so let's move on to the digits 6, 7, 8 and 9. Unfortunately in the range up to 40,000, no numbers satisfy but if we extend the range to one million, we find that for the digit 6, three numbers satisfy: 646661, 666181 and 766643 with factorisations as follows (permalink):

646661 = 61 * 10601 with total of 6

666181 = 61 * 67 * 163 with total of 6

766643 = 461 * 1663 with total of 6

For the digit 7, there are 10 numbers that qualify, namely 371777, 578977, 616777, 677777, 727679, 731773, 748177, 774769, 777767 and 777773. Their factorisations are as follows (permalink):

371777 = 7 * 173 * 307 with total of 7
578977 = 7 * 107 * 773 with total of 7
616777 = 7 * 17 * 71 * 73 with total of 7
677777 = 571 * 1187 with total of 7
727679 = 37 * 71 * 277 with total of 7
731773 = 7 * 107 * 977 with total of 7
748177 = 37 * 73 * 277 with total of 7
774769 = 277 * 2797 with total of 7
777767 = 17 * 45751 with total of 7
777773 = 709 * 1097 with total of 7

In the range up to one million, there are no numbers that satisfy for the digits 8 and 9.