cp's OEIS Frontend

A359169 Dirichlet inverse of the pointwise sum of A349905 (arithmetic derivative of prime shifted n) and A063524 (1, 0, 0, 0, ...).

%I A359169 #9 Dec 23 2022 17:26:25
%S A359169 1,-1,-1,-5,-1,-6,-1,-16,-9,-8,-1,-14,-1,-12,-10,-35,-1,-22,-1,-22,
%T A359169 -14,-14,-1,10,-13,-18,-56,-38,-1,-17,-1,-31,-16,-20,-16,42,-1,-24,
%U A359169 -20,-2,-1,-33,-1,-46,-54,-30,-1,243,-21,-46,-22,-62,-1,-10,-18,-26,-26,-32,-1,140,-1,-38,-86,160,-22,-41,-1,-70,-32,-53
%N A359169 Dirichlet inverse of the pointwise sum of A349905 (arithmetic derivative of prime shifted n) and A063524 (1, 0, 0, 0, ...).
%H A359169 Antti Karttunen, <a href="/A359169/b359169.txt">Table of n, a(n) for n = 1..10000</a>
%H A359169 Antti Karttunen, <a href="/A359169/a359169.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A359169 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A359169 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A349905(n/d) * a(d).
%F A359169 a(n) = A346241(A003961(n)).
%o A359169 (PARI)
%o A359169 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A359169 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A359169 A349905(n) = A003415(A003961(n));
%o A359169 memoA359169 = Map();
%o A359169 A359169(n) = if(1==n,1,my(v); if(mapisdefined(memoA359169,n,&v), v, v = -sumdiv(n,d,if(d<n,A349905(n/d)*A359169(d),0)); mapput(memoA359169,n,v); (v)));
%Y A359169 Cf. A003415, A003961, A346241, A349905.
%K A359169 sign
%O A359169 1,4
%A A359169 _Antti Karttunen_, Dec 23 2022