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A349905 Arithmetic derivative of A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

%I A349905 #17 Dec 01 2022 21:08:19
%S A349905 0,1,1,6,1,8,1,27,10,10,1,39,1,14,12,108,1,55,1,51,16,16,1,162,14,20,
%T A349905 75,75,1,71,1,405,18,22,18,240,1,26,22,216,1,103,1,87,95,32,1,621,22,
%U A349905 91,24,111,1,350,20,324,28,34,1,318,1,40,135,1458,24,119,1,123,34,131,1,945,1,44,119,147,24,151,1,837
%N A349905 Arithmetic derivative of A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).
%H A349905 Antti Karttunen, <a href="/A349905/b349905.txt">Table of n, a(n) for n = 1..20000</a>
%H A349905 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349905 a(n) = A003415(A003961(n)).
%t A349905 f1[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := d[s[n]]; Array[a, 100] (* _Amiram Eldar_, Dec 05 2021 *)
%o A349905 (PARI)
%o A349905 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349905 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349905 A349905(n) = A003415(A003961(n));
%Y A349905 Cf. A003415, A003961, A026424 (positions of odd terms), A028260 (of even terms), A066829 (parity of a(n)).
%Y A349905 Cf. also A343221, A343222, A347964, A347965, A348937, A353571.
%Y A349905 Cf. A358760, A358761, A358762, A358763 for indices of terms that of the form 4k+j, for j=0..3, and A358750, A358751, A358752, A358753 for their characteristic functions.
%K A349905 nonn
%O A349905 1,4
%A A349905 _Antti Karttunen_, Dec 05 2021