%I #14 Nov 19 2021 16:01:20

%S 0,1,1,7,1,12,1,33,10,16,1,68,1,20,18,131,1,87,1,96,22,28,1,296,16,32,

%T 67,124,1,167,1,473,30,40,26,449,1,44,34,428,1,215,1,180,147,52,1,

%U 1128,22,171,42,208,1,510,34,560,46,64,1,881,1,68,187,1611,38,311,1,264,54,295,1,1871,1,80,203,292,38,359

%N Dirichlet convolution of A003415 with A003959, where A003415 is the arithmetic derivative and A003959 is fully multiplicative with a(p) = (p+1).

%H Antti Karttunen, <a href="/A349173/b349173.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} A003415(d) * A003959(n/d).

%F a(n) = Sum_{d|n} A349133(d) * A349356(n/d). - _Antti Karttunen_, Nov 16 2021

%F For all n >= 1, a(n) >= A349133(n).

%t f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 09 2021 *)

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A349173(n) = sumdiv(n,d,A003415(d)*A003959(n/d));

%Y Cf. A003415, A003959, A349129, A349133, A349170, A349171, A349172, A349356.

%K nonn

%O 1,4

%A _Antti Karttunen_, Nov 09 2021