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A378526 Dirichlet inverse of A378548, where A378548 is the sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).

%I A378526 #9 Dec 01 2024 13:41:00
%S A378526 1,-2,-3,0,-5,6,-7,0,-1,10,-11,0,-13,14,14,0,-17,2,-19,0,20,22,-23,0,
%T A378526 -1,26,3,0,-29,-28,-31,0,32,34,34,0,-37,38,38,0,-41,-40,-43,0,8,46,
%U A378526 -47,0,-1,2,50,0,-53,-6,54,0,56,58,-59,0,-61,62,10,0,64,-64,-67,0,68,-68,-71,0,-73,74,8,0,76,-76,-79,0,0,82
%N A378526 Dirichlet inverse of A378548, where A378548 is the sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).
%C A378526 Agrees with A378525 on all odd n, and also on some even n: 2, 16, 32, 64, 96, 128, 160, 192, ...
%H A378526 Antti Karttunen, <a href="/A378526/b378526.txt">Table of n, a(n) for n = 1..20000</a>
%F A378526 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A378548(n/d) * a(d).
%F A378526 a(n) = Sum_{d|n} A023900(d)*A369454(n/d).
%F A378526 a(n) = Sum_{d|n} A055615(d)*A358777(n/d).
%o A378526 (PARI)
%o A378526 A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
%o A378526 A378548(n) = sumdiv(n,d,d*A353557(n/d));
%o A378526 memoA378526 = Map();
%o A378526 A378526(n) = if(1==n,1,my(v); if(mapisdefined(memoA378526,n,&v), v, v = -sumdiv(n,d,if(d<n,A378548(n/d)*A378526(d),0)); mapput(memoA378526,n,v); (v)));
%Y A378526 Cf. A023900, A055615, A353557, A358777, A369454, A378548.
%Y A378526 Cf. also A378525, A378527.
%K A378526 sign
%O A378526 1,2
%A A378526 _Antti Karttunen_, Dec 01 2024