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A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

%I A378435 #7 Nov 26 2024 17:14:21
%S A378435 1,3,4,4,6,12,8,6,7,18,12,16,14,24,24,9,18,21,20,24,32,36,24,24,16,42,
%T A378435 16,32,30,72,32,15,48,54,48,25,38,60,56,36,42,96,44,48,42,72,48,36,29,
%U A378435 48,72,56,54,48,72,48,80,90,60,96,62,96,56,24,84,144,68,72,96,144,72,33,74,114,64,80,96,168,80,54,34
%N A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.
%C A378435 The first negative term is a(2592) = -48.
%C A378435 Apparently differs from A325973 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...
%H A378435 Antti Karttunen, <a href="/A378435/b378435.txt">Table of n, a(n) for n = 1..20000</a>
%F A378435 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A378434(n/d) * a(d).
%o A378435 (PARI)
%o A378435 A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
%o A378435 A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
%o A378435 A378434(n) = ((A158523(n)+A178450(n))/2);
%o A378435 memoA378435 = Map();
%o A378435 A378435(n) = if(1==n,1,my(v); if(mapisdefined(memoA378435,n,&v), v, v = -sumdiv(n,d,if(d<n,A378434(n/d)*A378435(d),0)); mapput(memoA378435,n,v); (v)));
%Y A378435 Dirichlet inverse of A378434.
%Y A378435 Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433.
%K A378435 sign
%O A378435 1,2
%A A378435 _Antti Karttunen_, Nov 26 2024