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A347965 Dirichlet convolution of A003415 (arithmetic derivative) with A003961 (prime shift towards larger primes).

%I A347965 #9 Oct 15 2021 21:34:07
%S A347965 0,1,1,7,1,13,1,33,11,17,1,75,1,23,20,131,1,104,1,103,26,29,1,329,17,
%T A347965 35,82,145,1,196,1,473,32,41,30,552,1,47,38,461,1,274,1,187,181,57,1,
%U A347965 1259,25,194,44,229,1,682,36,659,50,65,1,1052,1,73,247,1611,42,352,1,271,60,366,1,2332,1,83,245,313,42
%N A347965 Dirichlet convolution of A003415 (arithmetic derivative) with A003961 (prime shift towards larger primes).
%H A347965 Antti Karttunen, <a href="/A347965/b347965.txt">Table of n, a(n) for n = 1..16384</a>
%H A347965 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A347965 a(n) = Sum_{d|n} A003415(d) * A003961(n/d).
%o A347965 (PARI)
%o A347965 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347965 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A347965 A347965(n) = sumdiv(n,d,A003415(n/d)*A003961(d));
%Y A347965 Cf. A003415, A003961.
%Y A347965 Cf. also A347130, A347964.
%K A347965 nonn
%O A347965 1,4
%A A347965 _Antti Karttunen_, Sep 25 2021