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

%I A345053 #22 Jul 06 2021 01:55:19
%S A345053 8,16,32,64,98,128,147,256,512,1024,1552,2048,2597,2752,3088,4064,
%T A345053 4096,4112,5648,6112,6176,7184,7399,8128,8192,8224,9232,9344,10256,
%U A345053 10768,12256,12304,14368,14864,16384,16448,17003,18448,18464,18688,19472,19984,20512,20992,22544,24512,24608,25616,27152,30224,31409,32272,32768
%N A345053 Positions of zeros in A345055, which is the Dirichlet inverse of A011772.
%H A345053 Chai Wah Wu, <a href="/A345053/b345053.txt">Table of n, a(n) for n = 1..10000</a>
%F A345053 From _Chai Wah Wu_, Jul 05 2021: (Start)
%F A345053 Theorem: 2^i for i >= 3 are terms.
%F A345053 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.
%F A345053 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.
%F A345053 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.
%F A345053 (End)
%F A345053 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
%o A345053 (PARI) isA345053(n) = (0==A345055(n));
%Y A345053 Cf. A011772, A345055.
%K A345053 nonn
%O A345053 1,1
%A A345053 _Antti Karttunen_, Jul 01 2021