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A002131
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Sum of divisors d of n such that n/d is odd.
(Formerly M0937 N0351)
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54
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1, 2, 4, 4, 6, 8, 8, 8, 13, 12, 12, 16, 14, 16, 24, 16, 18, 26, 20, 24, 32, 24, 24, 32, 31, 28, 40, 32, 30, 48, 32, 32, 48, 36, 48, 52, 38, 40, 56, 48, 42, 64, 44, 48, 78, 48, 48, 64, 57, 62, 72, 56, 54, 80, 72, 64, 80, 60, 60, 96, 62, 64, 104, 64, 84, 96, 68, 72, 96, 96, 72
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OFFSET
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1,2
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COMMENTS
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Glaisher calls this Delta'(n) or Delta'_1(n). - N. J. A. Sloane, Nov 24 2018
Cayley begins article 386 with "To find the value of A, = 8{q/(1-q)^2 + q^3/(1-q^3)^2 +&c.}," where A is 8 time the g.f. of this sequence. - Michael Somos, Aug 01 2011
a(n) = 2*(a(n-1) - a(n-4) + a(n-9) ... +- a(n-i^2) ...) up to the last positive number n - i^2, and if n is a square, then a(0) should be replaced by n/2 (cf. Halphen). - Michel Marcus, Oct 14 2012
a(n) is also the total number of odd parts in the partitions of n into equal parts.
a(n) = n iff n is a power of 2.
a(n) = n + 1 iff n is an odd prime. (End)
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REFERENCES
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A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 294, Art. 386.
G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(18). MR0121327 (22 #12066)
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eqs. (5.1.29.3), (5.1.29.9).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Expansion of K(k^2) * (K(k^2) - E(k^2)) / (2 * Pi^2) in powers of q where q is Jacobi's nome and K(), E() are complete elliptic integrals. - Michael Somos, Aug 01 2011
Multiplicative with a(p^e) = p^e if p = 2; (p^(e+1)-1)/(p-1) if p > 2. - David W. Wilson, Aug 01 2001
a(n) = sigma(n) - sigma(n/2) for even n and = sigma(n) otherwise where sigma(n) is the sum of divisors of n (A000203). - Valery A. Liskovets, Apr 07 2002
G.f.: A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = 2*u1*u6 - u1*u3 - 10*u2*u6 + u2^2 + 2*u2*u3 + 9*u6^2. - Michael Somos, Apr 10 2005
G.f.: A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 - 3*u6)^2 - (u1 - 2*u2) * (u3 - 2*u6). - Michael Somos, Sep 06 2005
G.f.: Sum_{k>0} x^(2*k - 1) / (1 - x^(2*k - 1))^2. - Michael Somos, Aug 17 2005
G.f.: (1/8) * theta_4''(0) / theta_4(0) = (Sum_{k>0} -(-1)^k * k^2 q^(k^2)) / (Sum_{k in Z} (-1)^k * q^(k^2)) where theta_4(u) is one of Jacobi's theta functions.
G.f.: A(q) = Z'(0) * K^2 / (2 * Pi^2) = (K - E) * K /(2 * Pi^2) where Z(u) is the Jacobi Zeta function and K, E are complete elliptic integrals. - Michael Somos, Sep 06 2005
Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 1/2^s). - Michael Somos, Apr 05 2003
a(n) = n * Sum_{c|n} 1/c, where c are odd numbers (A005408) dividing n. a(n) = A069359(n) + n. a(n) = A000035(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d) = Sum_{d|n} A000035(d) * A000027(n/d). -Jaroslav Krizek, Nov 07 2013
G.f.: A(x) = (1/2)*Sum_{n = -oo..oo} x^(2*n+1)/(1 - x^(2*n+1))^2.
A(x) = Sum_{n = -oo..oo} x^(4*n+1)/(1 - x^(4*n+1))^2.
a(2*n) = 2*a(n); a(2*n+1) = A008438(n). (End)
Expansion of (-1/2) x (d phi(-x) / dx) / phi(-x) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jul 01 2023
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EXAMPLE
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G.f. = q + 2*q^2 + 4*q^3 + 4*q^4 + 6*q^5 + 8*q^6 + 8*q^7 + 8*q^8 + 13*q^9 + ...
The divisors of 6 are 1, 2, 3, and 6. Only 6/2 and 6/6 are odd. Hence, a(6) = 2 + 6 = 8.
As 120 = 15 * 2^3 where 15 is odd and 2^3 is the largest power of 2 dividing 120, a(120) = sigma(15) * 2^3 = 24 * 8 = 192. - David A. Corneth, Aug 12 2019
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1]. There are 8 odd parts, so a(6) = 8. - Omar E. Pol, Nov 26 2019
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MAPLE
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a:= proc(n) local e;
e:= 2^padic:-ordp(n, 2);
e*numtheory:-sigma(n/e)
end proc:
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MATHEMATICA
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a[n_]:=Total[Cases[Divisors[n], d_ /; OddQ[n/d]]]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Mar 18 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, # / GCD[#, 2] &]] (* Michael Somos, Aug 01 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1/8) EllipticK[ m] ( EllipticK[ m] - EllipticE[ m] ) / (Pi/2 )^2, {q, 0, n}]] (* Michael Somos, Aug 01 2011 *)
Table[Total[Select[Divisors[n], OddQ[n/#]&]], {n, 80}] (* Harvey P. Dale, Jun 05 2015 *)
a[ n_] := SeriesCoefficient[ With[ {m = InverseEllipticNomeQ[q]}, (1/2) (EllipticK[ m] / Pi)^2 (D[ JacobiZeta[ JacobiAmplitude[x, m], m], x] /. x -> 0)], {q, 0, n}]; (* Michael Somos, Mar 17 2017 *)
f[2, e_] := 2^e; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - (p<3) * X) / ((1 - X) * (1 - p*X))) [n])}; /* Michael Somos, Apr 05 2003 */
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d / gcd(d, 2)))}; /* Michael Somos, Apr 05 2003 */
(PARI) a(n) = my(v = valuation(n, 2)); sigma(n>>v)<<v \\ David A. Corneth, Aug 12 2019
(Haskell)
a002131 n = sum [d | d <- [1..n], mod n d == 0, odd $ div n d]
(Magma) [&+[d:d in Divisors(m)|IsOdd(Floor(m/d))] :m in [1..75]]; // Marius A. Burtea, Aug 12 2019
(Python)
from math import prod
from sympy import factorint
def A002131(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Dec 17 2021
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CROSSREFS
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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STATUS
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approved
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