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
Examples
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.
Links
- Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
- Omar E. Pol, Comments on A245579.
Crossrefs
Programs
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Maple
seq(n*numtheory:-tau(n/2^padic:-ordp(n,2)), n=1..100); # Robert Israel, Apr 26 2017
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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 *) -
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
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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
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)
Extensions
Edited by N. J. A. Sloane, Apr 27 2022