# [seqfan] Sequences about product or sum of digits of n

David Corneth davidacorneth at gmail.com
Mon Jun 6 20:16:59 CEST 2016

```Hi all,

In OEIS there are sequences using the product of digits of n or the sum of
digits of n.
Some of those sequence don't really look at n itself. Such sequences are,
for example (there are many more):

A038367: Numbers n with property that (product of digits of n) is divisible
by (sum of digits of n).
A055931: Product of the digits of n divides the sum of the digits of n.
A061013: Numbers n such that (product of digits of n) is divisible by (sum
of digits of n), where 0's are not permitted.
A062996: Sum of digits is greater than or equal to product of digits.
A062997: Sum of digits is strictly greater than product of digits.
A117720: Numbers for which the sum of the digits is the square root of the
product of their digits.

I think there is an opportunity to look describe such sequences in another
way.
Looking at A061013, we see that we can take a term and we can permute all
digits to create new terms. Eg. 246 is a term, so 264, 426, 462, 624 and
642 are as well. The sequence is about the sum and product of digits. Their
order doesn't matter as both multiplication and addition are commutative.
We could therefore pick one, e.g. the term with digits in nonincreasing
order, i.e. a term from A009994.

If we look a little closer to A038367, one must be a bit more careful
permuting the digits. As terms can contain zero's, placing them up front
would mean they don't become part of that number. For example, 102 is an
element of A038367, so is and 120, 201, 210. But not necessarily 012 and
021 as they're actually 12 and 21. For such sequences with terms having a
digit 0, we can use A179239; Permutation classes of integers, each
identified by its smallest member.

>From examples of sequences I gave, A055931, A061013 and A117720 have no 0's
(except A117720 has 0 itself so would have 0 prepended to it) we can have
the intersection of A009994 and respectively one of these sequenes.
A038367, A062996 and A062997 have 0's in them. The intersection of A179239
and respectively one of those sequences would give the sequences I suggest.
None of these intersections are in OEIS.

I think describing these sequences like that makes it easier to look for
terms. The density of both  A009994 and A179239 is zero so one effictively
looks at less terms when looking upto some bound. I already put a
next-function in A179239; given a term of A179239(n), find A179239(n+1).
Also, each term gives more insight to others. Now we just have to put 246
to know that all of 264, 426, 462, 624 and 642 are as well.

What do you think of this approach?

Best,
David
```