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

1,1

Links

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

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

Prog

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

Crossrefs

Cf. A011772, A345055.

Sequence in context: A277128 A054743 A192135 * A256817 A281016 A048169

Adjacent sequences: A345050 A345051 A345052 * A345054 A345055 A345056

Keyword

nonn

Author

Antti Karttunen, Jul 01 2021

Status

approved