A349124 - OEIS

%I #11 Nov 15 2021 03:40:22

%S 0,1,1,4,1,15,1,10,4,21,1,48,1,27,24,20,1,42,1,72,30,39,1,110,4,45,10,

%T 96,1,279,1,35,42,57,36,120,1,63,48,170,1,369,1,144,78,75,1,210,4,54,

%U 60,168,1,90,48,230,66,93,1,828,1,99,102,56,54,549,1,216,78,531,1,260,1,117,66,240,54,639,1,330,20,129

%N a(n) = A349123(n) / A003557(n), where A349123 is the Dirichlet convolution of the arithmetic derivative with n*tau(n).

%H Antti Karttunen, <a href="/A349124/b349124.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A349123(n) / A003557(n).

%t f[p_, e_] := p^(e - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &] / s[n]; Array[a, 100] (* _Amiram Eldar_, Nov 08 2021 *)

%o (PARI)

%o A003557(n) = (n/factorback(factorint(n)[, 1]));

%o A349124(n) = (A349123(n) / A003557(n)); \\ Needs also code from A349123.

%Y Cf. A003415, A003557, A038040, A349123.

%Y Cf. also A347129.

%K nonn

%O 1,4

%A _Antti Karttunen_, Nov 08 2021