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A245579 Number of odd divisors of n multiplied by n. 25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sum of all parts of all partitions of n into consecutive parts. - Omar E. Pol, Apr 23 2017

Sum of all parts of all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 05 2017

Row sums of A299765. - Omar E. Pol, Jul 23 2018

Row sums of A328362. - Omar E. Pol, Oct 20 2019

Row sums of A285891 and of A328365. - Omar E. Pol, Nov 04 2019

Number of partitions of n into consecutive parts, multiplied by n. Also, number of partitions of n into an odd number of equal parts, multiplied by n. - Omar E. Pol, Nov 05 2019

LINKS

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

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.

a(n) = n*A000005(n)/A001511(n) = A038040(n)/A001511(n). - Omar E. Pol, Apr 24 2018

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. On the other hand, for n = 10 there are two partitions of 10 into consecutive integers: [10] and [4, 3, 2, 1], and the sum of all parts of these partitions is 10 + 4 + 3 + 2 + 1 = 20, so a(10) = 20. - Omar E. Pol, Apr 23 2017

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

CROSSREFS

Cf. A000005, A001227, A001511, A038040, A285891, A299765, A328362, A328365.

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

STATUS

approved

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Last modified June 21 06:24 EDT 2021. Contains 345358 sequences. (Running on oeis4.)