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A349359 Sum of A064216 and its Dirichlet inverse, where A064216 = A064989(2n-1), and A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

%I A349359 #10 Nov 25 2021 20:35:12
%S A349359 2,0,0,4,0,12,0,12,9,16,0,22,0,44,24,5,0,40,0,60,66,40,0,14,16,36,51,
%T A349359 10,0,106,0,82,60,56,88,26,0,124,54,-10,0,-46,0,144,134,48,0,235,121,
%U A349359 140,84,86,0,19,80,-108,186,136,0,-44,0,236,211,29,72,158,0,216,72,62,0,152,0,284,190,10,220,98,0,260,181
%N A349359 Sum of A064216 and its Dirichlet inverse, where A064216 = A064989(2n-1), and A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.
%C A349359 Compare to A323894 which in contrast to this sequence seems to have only nonnegative terms.
%H A349359 Antti Karttunen, <a href="/A349359/b349359.txt">Table of n, a(n) for n = 1..16384</a>
%H A349359 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349359 a(n) = A064216(n) + A349358(n).
%F A349359 a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1<d<n} A064216(d) * A349358(n/d).
%o A349359 (PARI)
%o A349359 A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
%o A349359 A064216(n) = A064989((2*n)-1);
%o A349359 memoA349358 = Map();
%o A349359 A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(d<n,A064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
%o A349359 A349359(n) = (A064216(n)+A349358(n));
%Y A349359 Cf. A064216, A064989, A349358.
%Y A349359 Cf. also A323894, A349126.
%K A349359 sign
%O A349359 1,1
%A A349359 _Antti Karttunen_, Nov 17 2021