cp's OEIS Frontend

A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).

%I A329643 #12 Nov 22 2019 18:54:16
%S A329643 0,0,0,1,0,2,0,2,1,2,0,6,0,2,2,6,0,4,0,7,2,2,0,16,1,2,0,18,0,11,0,6,2,
%T A329643 2,2,22,0,2,2,24,0,17,0,20,2,2,0,28,1,1,2,48,0,16,2,28,2,2,0,39,0,2,
%U A329643 -3,30,2,36,0,84,2,19,0,36,0,2,-2,258,2,38,0,28,4,2,0,69,2,2,2,72,0,31,2,228,2,2,2,76,0,4,14,37,0,94,0,136,-3
%N A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * A323244(d), where A323244(x) gives the deficiency of A156552(x).
%H A329643 Antti Karttunen, <a href="/A329643/b329643.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H A329643 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A329643 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A329643 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A329643 a(n) = Sum_{d|n} [-1 == A008683(n/d)] * (2*A156552(d) - A323243(d)).
%F A329643 a(n) = A329642(n) - A329644(n).
%F A329643 For all n, a(A000040(n)) = 0, a(A006881(n)) = 2.
%o A329643 (PARI)
%o A329643 A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
%o A329643 A323243(n) = if(1==n,0,sigma(A156552(n)));
%o A329643 A329643(n) = sumdiv(n,d,(-1==moebius(n/d))*((2*A156552(d))-A323243(d)));
%Y A329643 Cf. A006881, A008683, A156552, A323243, A323244, A329642, A329644.
%Y A329643 Cf. A329646 (inverse Möbius transform).
%K A329643 sign
%O A329643 1,6
%A A329643 _Antti Karttunen_, Nov 21 2019