A329642 - OEIS

%I #9 Nov 22 2019 18:54:07

%S 0,1,1,2,1,4,1,6,0,5,1,11,1,16,2,6,1,13,1,19,-3,18,1,24,-4,46,4,23,1,

%T 13,1,30,14,82,-2,18,1,256,-12,24,1,39,1,67,9,226,1,52,-12,20,8,131,1,

%U 28,-19,72,-12,748,1,53,1,1362,17,22,10,45,1,215,254,43,1,48,1,3838,11,407,-10,109,1,84,-12,5458,1,48,-72,12250,-348,32,1,18

%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="/A329642/b329642.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) = A329643(n) + A329644(n).

%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 A329642(n) = sumdiv(n,d,(1==moebius(n/d))*((2*A156552(d))-A323243(d)));

%Y Cf. A008683, A156552, A323243, A323244, A329643, A329644.

%Y Cf. A329645 (inverse Möbius transform).

%K sign

%O 1,4

%A _Antti Karttunen_, Nov 21 2019