A245579 Number of odd divisors of n multiplied by n.

1, 2, 6, 4, 10, 12, 14, 8, 27, 20, 22, 24, 26, 28, 60, 16, 34, 54, 38, 40, 84, 44, 46, 48, 75, 52, 108, 56, 58, 120, 62, 32, 132, 68, 140, 108, 74, 76, 156, 80, 82, 168, 86, 88, 270, 92, 94, 96, 147, 150, 204, 104, 106, 216, 220, 112, 228, 116, 118, 240, 122

Offset

1,2

Links

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Omar E. Pol, Comments on A245579.

Formula

a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = p^e * (e+1) if p>2.

a(n) = n * A001227(n).

G.f.: Sum_{k>0 odd} k * x^k / (1 - x^k)^2.

From Amiram Eldar, Dec 31 2022: (Start)

Dirichlet g.f.: zeta(s-1)^2*(1-1/2^(s-1)).

Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma + 2*log(2) - 1)*n^2/8, where gamma is Euler's constant (A001620). (End)

Example

G.f. = x + 2*x^2 + 6*x^3 + 4*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 8*x^8 + ...
For n = 10 there are two odd divisors of 10: 1 and 5, so a(10) = 2*10 = 20.

Maple

seq(n*numtheory:-tau(n/2^padic:-ordp(n, 2)), n=1..100); # Robert Israel, Apr 26 2017

Mathematica

a[ n_] := If[ n < 1, 0, n Sum[ Mod[d, 2], {d, Divisors @ n}]];
(* Second program: *)
Table[n DivisorSum[n, 1 &, OddQ], {n, 61}] (* Michael De Vlieger, Apr 24 2017 *)

Prog

(PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, d%2))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, if( k%2, k * x^k / (1 - x^k)^2), x * O(x^n)), n))};
(PARI) {a(n) = if( n<1, 0, n * numdiv(n / 2^valuation(n, 2)))} \\ Fast when n has many divisors. Jens Kruse Andersen, Jul 26 2014
(Python)
from sympy import divisors
def a(n): return n*len(list(filter(lambda i: i%2==1, divisors(n)))) # Indranil Ghosh, Apr 24 2017
(Python)
from math import prod
from sympy import factorint
def A245579(n): return n*prod(e+1 for e in factorint(n>>(~n&n-1).bit_length()).values()) # Chai Wah Wu, Dec 31 2023

Crossrefs

Cf. A000005, A001227, A001511, A001620, A038040, A285891, A299765, A328362, A328365, A352257, A352505.

Sequence in context: A119018 A264647 A094748 * A245788 A065879 A065880

Adjacent sequences: A245576 A245577 A245578 * A245580 A245581 A245582

Keyword

nonn,mult

Author

Michael Somos, Jul 26 2014

Extensions

Edited by N. J. A. Sloane, Apr 27 2022

Status

approved