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A376414 Dirichlet inverse of sigma(n)-A003415(n), where sigma is the sum of divisors function and A003415 is the arithmetic derivative.

%I A376414 #6 Nov 15 2024 15:12:06
%S A376414 1,-2,-3,1,-5,5,-7,1,2,9,-11,-2,-13,13,14,2,-17,-2,-19,-4,20,21,-23,
%T A376414 -4,4,25,2,-6,-29,-21,-31,5,32,33,34,3,-37,37,38,-6,-41,-31,-43,-10,
%U A376414 -8,45,-47,-9,6,-4,50,-12,-53,-6,54,-8,56,57,-59,8,-61,61,-12,13,64,-51,-67,-16,68,-57,-71,6,-73,73,-8,-18
%N A376414 Dirichlet inverse of sigma(n)-A003415(n), where sigma is the sum of divisors function and A003415 is the arithmetic derivative.
%H A376414 Antti Karttunen, <a href="/A376414/b376414.txt">Table of n, a(n) for n = 1..20000</a>
%F A376414 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A343224(n/d) * a(d).
%o A376414 (PARI)
%o A376414 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A376414 A343224(n) = (sigma(n) - A003415(n));
%o A376414 memoA376414 = Map();
%o A376414 A376414(n) = if(1==n,1,my(v); if(mapisdefined(memoA376414,n,&v), v, v = -sumdiv(n,d,if(d<n,A343224(n/d)*A376414(d),0)); mapput(memoA376414,n,v); (v)));
%Y A376414 Dirichlet inverse of A343224.
%Y A376414 Cf. A000203, A003415.
%K A376414 sign
%O A376414 1,2
%A A376414 _Antti Karttunen_, Nov 15 2024