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A359791 Dirichlet inverse of function f(n) = 1 + A349905(n), where A349905(n) is the arithmetic derivative of prime shifted n.

%I A359791 #7 Jan 17 2023 10:01:28
%S A359791 1,-2,-2,-3,-2,-1,-2,-8,-7,-3,-2,0,-2,-7,-5,-16,-2,0,-2,-4,-9,-9,-2,
%T A359791 23,-11,-13,-40,-12,-2,12,-2,-16,-11,-15,-11,42,-2,-19,-15,21,-2,12,
%U A359791 -2,-16,-24,-25,-2,128,-19,-12,-17,-24,-2,67,-13,17,-21,-27,-2,105,-2,-33,-48,48,-17,12,-2,-28,-27,0,-2,224
%N A359791 Dirichlet inverse of function f(n) = 1 + A349905(n), where A349905(n) is the arithmetic derivative of prime shifted n.
%H A359791 Antti Karttunen, <a href="/A359791/b359791.txt">Table of n, a(n) for n = 1..16384</a>
%H A359791 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A359791 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A349905(n/d)) * a(d).
%F A359791 a(n) = A359790(A003961(n)).
%o A359791 (PARI)
%o A359791 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A359791 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A359791 A349905(n) = A003415(A003961(n));
%o A359791 memoA359791 = Map();
%o A359791 A359791(n) = if(1==n,1,my(v); if(mapisdefined(memoA359791,n,&v), v, v = -sumdiv(n,d,if(d<n,(1+A349905(n/d))*A359791(d),0)); mapput(memoA359791,n,v); (v)));
%Y A359791 Cf. A003415, A003961, A349905, A359790.
%Y A359791 Cf. A359764 (parity of terms), A359765 (positions of odd terms), A359766 (of even terms).
%Y A359791 Cf. also A359169.
%K A359791 sign
%O A359791 1,2
%A A359791 _Antti Karttunen_, Jan 13 2023