[seqfan] Re: Do Mersenne primes ever divide A076078(n)?

[Max Alekseyev]

On Tue, Oct 12, 2010 at 11:38 PM,  <ghodges14 at comcast.net> wrote:

> A097416 contains the first 31 distinct values of A076078(n) which are not powers of 2
> (cf. http://oeis.org/classic/A097416).  11 larger values can be found in A076078's subsequences A097211, A097215, and, if I understand the inverse binomial transform correctly, A000371 (http://oeis.org/classic/A000371).
>
> According to Dario Alpern's factorization calculator, none of those 42 values is a multiple of 3, 7, 31, or any larger Mersenne prime. Since the formula for A076078(n) is based on Mersenne numbers (A000225), the natural conjecture is that no Mersenne prime divides any member of A076078(n).  Is this known or provable (or perhaps disprovable)?

This is true for 3 and 7 but not for 31.
In particular, A076078(2^3*3^3*5^3*7*11*13*19*23) and
A076078(2*3*5*7*11*13*19*23^3) are both divisible by 31.

> It also appears that, for any n, A076078(n) is congruent to A008836(n) mod 3.

This is true.

> I don't see any similar patterns for larger Mersenne primes.

Modulo 7, we have need to count
k1 = the number of primes p dividing n, whose exponent (i.e.,
valuation of n w.r.t. p) is == 1 (mod 3)
k2 = the number of primes dividing n, whose exponent is == 2 (mod 3)
then
A076078(n) == -(-2)^k1 (mod 7), if k1>0
A076078(n) == 2*(-1)^k2 - 1 (mod 7), if k1=0

It is easy to see that these expressions are never 0 modulo 7.

Regards,
Max