A329643 - OEIS

%I #12 Nov 22 2019 18:54:16

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

%H 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 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = Sum_{d|n} [-1 == A008683(n/d)] * (2*A156552(d) - A323243(d)).

%F a(n) = A329642(n) - A329644(n).

%F For all n, a(A000040(n)) = 0, a(A006881(n)) = 2.

%o (PARI)

%o 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 A323243(n) = if(1==n,0,sigma(A156552(n)));

%o A329643(n) = sumdiv(n,d,(-1==moebius(n/d))*((2*A156552(d))-A323243(d)));

%Y Cf. A006881, A008683, A156552, A323243, A323244, A329642, A329644.

%Y Cf. A329646 (inverse Möbius transform).

%K sign

%O 1,6

%A _Antti Karttunen_, Nov 21 2019