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A364574 Dirichlet inverse of A005941.

%I A364574 #9 Aug 05 2023 15:46:43
%S A364574 1,-2,-3,0,-5,6,-9,0,2,10,-17,0,-33,18,19,0,-65,-4,-129,0,35,34,-257,
%T A364574 0,12,66,0,0,-513,-38,-1025,0,67,130,69,0,-2049,258,131,0,-4097,-70,
%U A364574 -8193,0,-22,514,-16385,0,56,-24,259,0,-32769,0,133,0,515,1026,-65537,0,-131073,2050,-42,0,261,-134,-262145,0,1027
%N A364574 Dirichlet inverse of A005941.
%H A364574 Antti Karttunen, <a href="/A364574/b364574.txt">Table of n, a(n) for n = 1..8192</a>
%F A364574 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A005941(n/d) * a(d).
%F A364574 a(p) = -A005941(p) for all primes p.
%o A364574 (PARI)
%o A364574 A005941(n) = { my(f=factor(n), p, 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]); (1+res) }; \\ (After _David A. Corneth_'s program for A156552)
%o A364574 memoA364574 = Map();
%o A364574 A364574(n) = if(1==n,1,my(v); if(mapisdefined(memoA364574,n,&v), v, v = -sumdiv(n,d,if(d<n,A005941(n/d)*A364574(d),0)); mapput(memoA364574,n,v); (v)));
%Y A364574 Cf. A005941, A364575.
%Y A364574 Cf. also A324640.
%K A364574 sign
%O A364574 1,2
%A A364574 _Antti Karttunen_, Aug 05 2023