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

Charles Greathouse charles.greathouse at case.edu
Mon Jun 27 20:35:45 CEST 2016

```I think this is often the best approach (using the most natural quotient
sequence).

On Monday, June 6, 2016, David Corneth <davidacorneth at gmail.com> wrote:

> 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
>
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> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Charles Greathouse
Case Western Reserve University
```