login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.
7

%I #16 Oct 19 2021 14:21:27

%S 0,1,1,7,1,12,1,30,10,16,1,65,1,20,18,104,1,83,1,93,22,28,1,254,16,32,

%T 63,121,1,167,1,320,30,40,26,391,1,44,34,374,1,215,1,177,143,52,1,840,

%U 22,165,42,205,1,450,34,494,46,64,1,827,1,68,183,912,38,311,1,261,54,295,1,1430,1,80,197,289,38,359

%N a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

%C Dirichlet convolution of Dedekind psi function (A001615) with the arithmetic derivative (A003415).

%H Antti Karttunen, <a href="/A347132/b347132.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A347132/a347132.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = Sum_{d|n} A001615(n/d) * A003415(d).

%t Table[DivisorSum[n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/#]*If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &], {n, 78}] (* _Michael De Vlieger_, Oct 19 2021, after _Jan Mangaldan_ at A001615 *)

%o (PARI)

%o A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615

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

%o A347132(n) = sumdiv(n,d,A001615(n/d)*A003415(d));

%Y Cf. A001615, A003415.

%Y Cf. also A305809, A322577, A347131, A347133, A347135.

%K nonn

%O 1,4

%A _Antti Karttunen_, Aug 23 2021