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A245579 Number of odd divisors of n multiplied by n.

%I A245579 #68 Dec 31 2023 11:30:27
%S A245579 1,2,6,4,10,12,14,8,27,20,22,24,26,28,60,16,34,54,38,40,84,44,46,48,
%T A245579 75,52,108,56,58,120,62,32,132,68,140,108,74,76,156,80,82,168,86,88,
%U A245579 270,92,94,96,147,150,204,104,106,216,220,112,228,116,118,240,122
%N A245579 Number of odd divisors of n multiplied by n.
%H A245579 Jens Kruse Andersen, <a href="/A245579/b245579.txt">Table of n, a(n) for n = 1..10000</a>
%H A245579 Omar E. Pol, <a href="/A245579/a245579.txt">Comments on A245579</a>.
%F A245579 a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = p^e * (e+1) if p>2.
%F A245579 a(n) = n * A001227(n).
%F A245579 G.f.: Sum_{k>0 odd} k * x^k / (1 - x^k)^2.
%F A245579 From _Amiram Eldar_, Dec 31 2022: (Start)
%F A245579 Dirichlet g.f.: zeta(s-1)^2*(1-1/2^(s-1)).
%F A245579 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)
%e A245579 G.f. = x + 2*x^2 + 6*x^3 + 4*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 8*x^8 + ...
%e A245579 For n = 10 there are two odd divisors of 10: 1 and 5, so a(10) = 2*10 = 20.
%p A245579 seq(n*numtheory:-tau(n/2^padic:-ordp(n,2)), n=1..100); # _Robert Israel_, Apr 26 2017
%t A245579 a[ n_] := If[ n < 1, 0, n Sum[ Mod[d, 2], {d, Divisors @ n}]];
%t A245579 (* Second program: *)
%t A245579 Table[n DivisorSum[n, 1 &, OddQ], {n, 61}] (* _Michael De Vlieger_, Apr 24 2017 *)
%o A245579 (PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, d%2))};
%o A245579 (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))};
%o A245579 (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
%o A245579 (Python)
%o A245579 from sympy import divisors
%o A245579 def a(n): return n*len(list(filter(lambda i: i%2==1, divisors(n)))) # _Indranil Ghosh_, Apr 24 2017
%o A245579 (Python)
%o A245579 from math import prod
%o A245579 from sympy import factorint
%o A245579 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
%Y A245579 Cf. A000005, A001227, A001511, A001620, A038040, A285891, A299765, A328362, A328365, A352257, A352505.
%K A245579 nonn,mult
%O A245579 1,2
%A A245579 _Michael Somos_, Jul 26 2014
%E A245579 Edited by _N. J. A. Sloane_, Apr 27 2022