A345053 Positions of zeros in A345055, which is the Dirichlet inverse of A011772.

8, 16, 32, 64, 98, 128, 147, 256, 512, 1024, 1552, 2048, 2597, 2752, 3088, 4064, 4096, 4112, 5648, 6112, 6176, 7184, 7399, 8128, 8192, 8224, 9232, 9344, 10256, 10768, 12256, 12304, 14368, 14864, 16384, 16448, 17003, 18448, 18464, 18688, 19472, 19984, 20512, 20992, 22544, 24512, 24608, 25616, 27152, 30224, 31409, 32272, 32768

Offset: 1

Author

Antti Karttunen, Jul 01 2021

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A011772, A345055.

Programs

PARI

isA345053(n) = (0==A345055(n));

Formula

From Chai Wah Wu, Jul 05 2021: (Start)

Theorem: 2^i for i >= 3 are terms.

Proof: This can be shown by induction on i. For the inductive step, A345055(1)=1, A345055(2)=-3, A345055(3)=2, and A011772(2^i)=2^(i+1)-1.

So for the divisors 1,2,4 for 2^i, A011772(2^i)*A345055(1)+A011772(2^(i-1))*A345055(2)+A011772(2^(i-2))*A345055(4)=0.

A345055(d)=0 for the other proper divisors d of 2^i by the inductive hypothesis as d is a power of 2 and this implies A345033(2^i)=0 for i>=3.

(End)

Conjecture: all terms are of the form 2^i, 2^i*p, 2^i*p*q or 7^2*p for some primes p and q. - Chai Wah Wu, Jul 05 2021