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

ghodges14 at comcast.net ghodges14 at comcast.net
Wed Oct 13 05:38:26 CEST 2010


Hello SeqFans,

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)?  

It also appears that, for any n, A076078(n) is congruent to A008836(n) mod 3.  Since A008836(n) is always 1 or -1, this would imply that 3 never divides A076078(n). I don't see any similar patterns for larger Mersenne primes.   

Matt Vandermast

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