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A378753 Dirichlet inverse of A378752, where A378752(n) = 2*sigma(n) - sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

%I A378753 #6 Dec 11 2024 20:17:55
%S A378753 1,-2,-2,3,-4,8,-4,6,9,12,-10,2,-10,16,16,24,-16,2,-16,-4,24,24,-18,
%T A378753 32,11,28,48,4,-28,-32,-26,96,28,36,32,83,-34,40,36,20,-40,-32,-40,
%U A378753 -22,4,48,-42,152,35,-2,40,-14,-48,80,48,64,48,60,-58,80,-56,64,44,384,56,-80,-64,-40,60,-64,-70,370,-68,76,18
%N A378753 Dirichlet inverse of A378752, where A378752(n) = 2*sigma(n) - sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).
%H A378753 Antti Karttunen, <a href="/A378753/b378753.txt">Table of n, a(n) for n = 1..20000</a>
%H A378753 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A378753 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A378753 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A378752(n/d) * a(d).
%o A378753 (PARI)
%o A378753 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378753 A378752(n) = ((2*sigma(n)) - sigma(A003961(n)));
%o A378753 memoA378753 = Map();
%o A378753 A378753(n) = if(1==n,1,my(v); if(mapisdefined(memoA378753,n,&v), v, v = -sumdiv(n,d,if(d<n,A378752(n/d)*A378753(d),0)); mapput(memoA378753,n,v); (v)));
%Y A378753 Cf. A000203, A003973, A378752 (Dirichlet inverse).
%K A378753 sign
%O A378753 1,2
%A A378753 _Antti Karttunen_, Dec 11 2024