A390258 - OEIS

%I #13 Nov 25 2025 12:41:02

%S 1,5,8,18,14,40,20,56,45,70,32,144,38,100,112,160,50,225,56,252,160,

%T 160,68,448,135,190,216,360,86,560,92,432,256,250,280,810,110,280,304,

%U 784,122,800,128,576,630,340,140,1280,273,675,400,684,158,1080,448,1120,448,430,176,2016

%N Multiplicative sequence a(n) with a(p^e) = p^(e-1) * (e + 1) * (p * (2 + e) - e) / 2 for prime p and e > 0.

%H Andrew Howroyd, <a href="/A390258/b390258.txt">Table of n, a(n) for n = 1..10000</a>

%F Dirichlet g.f.: (zeta(s-1))^3 / zeta(s).

%F Dirichlet convolution of A000027 and A018804.

%F Dirichlet convolution of A038040 and A000010.

%t f[p_, e_] := p^(e-1) * (e+1) * (p*(e+2) - e)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 30 2025 *)

%o (PARI) a(n) = { my(f = factor(n)); prod(i=1,#f~,f[i,1]^(f[i,2]-1)*(f[i,2]+1)*(f[i,1]*(2+f[i,2])-f[i,2])/2) }

%Y Cf. A000010, A000027, A018804, A038040.

%K nonn,easy,mult

%O 1,2

%A _Werner Schulte_, Oct 30 2025