A340774 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * zeta(2*s-1).

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 28, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 56, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 84, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 120, 65, 66, 67

Offset: 1

Author

Werner Schulte, Jan 20 2021

Keywords

• nonn
• easy
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000010, A069290, A055615, A037213.

Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k =

0..10: A046951 (k=0), this sequence (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

Programs

Maple

a:= n-> mul((i[1]^(i[2]+1)-i[1]^iquo(i[2]+1, 2))/(i[1]-1), i=ifactors(n)[2]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 20 2021

Mathematica

f[p_, e_] := (p^(e + 1) - p^Floor[(e + 1)/2])/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2021 *)

PARI

A340774(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((p^(e+1)-(p^((e+1)\2))) / (p-1))); }; \\ Antti Karttunen, Aug 19 2021

Formula

Multiplicative with a(p^e) = (p^(e+1)-p^floor((e+1)/2))/(p-1).

Dirichlet convolution of A000010 and A069290.

Dirichlet convolution with A055615 equals A037213.

G.f.: Sum_{k>=1} k * x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021

Sum_{k=1..n} a(k) ~ zeta(3)*n^2/2. - Vaclav Kotesovec, Aug 19 2021

a(n) = n * Sum_{d^2|n} 1/d. - Wesley Ivan Hurt, Feb 14 2022