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A245579
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Number of odd divisors of n multiplied by n.
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30
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = p^e * (e+1) if p>2.
G.f.: Sum_{k>0 odd} k * x^k / (1 - x^k)^2.
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)
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EXAMPLE
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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.
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MAPLE
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seq(n*numtheory:-tau(n/2^padic:-ordp(n, 2)), n=1..100); # Robert Israel, Apr 26 2017
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, n Sum[ Mod[d, 2], {d, Divisors @ n}]];
(* Second program: *)
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PROG
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(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
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CROSSREFS
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Cf. A000005, A001227, A001511, A001620, A038040, A285891, A299765, A328362, A328365, A352257, A352505.
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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