A002173 a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2. (Formerly M4467 N1895)

1, 1, -8, 1, 26, -8, -48, 1, 73, 26, -120, -8, 170, -48, -208, 1, 290, 73, -360, 26, 384, -120, -528, -8, 651, 170, -656, -48, 842, -208, -960, 1, 960, 290, -1248, 73, 1370, -360, -1360, 26, 1682, 384, -1848, -120, 1898, -528, -2208, -8, 2353, 651, -2320, 170

Offset

1,3

Comments

Multiplicative because it is the Inverse Moebius transform of [1, 0, -3^2, 0, 5^2, 0, -7^2, ...], which is multiplicative. - Christian G. Bower, May 18 2005

References

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

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).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.

J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

Frazer Jarvis and Helena A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J. (22) (2010), 171.

Jan Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.

Index entries for sequences mentioned by Glaisher.

Formula

Multiplicative with a(p^e) = 1 if p = 2; ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4); ((-p^2)^(e+1)-1)/(-p^2-1) if p == 3 (mod 4). - David W. Wilson, Sep 01 2001 [This can be written as a single formula: a(p^e) = ((p^2*Chi(p))^(e+1) - 1)/(p^2*Chi(p) - 1), Chi = A101455. - Jianing Song, Oct 30 2019]

G.f.: Sum_{n>=1} A056594(n-1)*n^2*q^n/(1-q^n).

Expansion of (1 - theta_4(q)^2 * theta_4(q^2)^4)/4 in powers of q. - Michael Somos, Aug 09 2006

Expansion of (1-eta(q)^4*eta(q^2)^6/eta(q^4)^4)/4 in powers of q.

G.f.: q*G'(q)/G(q), with G(q) = Product_{n>=1} (1-q^n)^(4n*A056594(n+1)).

a(n) = Sum_{d|n} d^2*sin(d*Pi/2). - Ridouane Oudra, Feb 21 2023

Example

The divisors of 15 are 1,3,5,15, so a(15)=(1^2+5^2)-(3^2+15^2) = -208.
G.f. = x + x^2 - 8*x^3 + x^4 + 26*x^5 - 8*x^6 - 48*x^7 + x^8 + 73*x^9 + ... - Michael Somos, Jun 25 2019

Maple

with(numtheory):
A002173:= proc(n)
    local count1, count3, d;
    count1 := 0:
    count3 := 0:
    for d in numtheory[divisors](n) do
        if d mod 4 = 1 then
            count1 := count1+d^2
        elif d mod 4 = 3 then
            count3 := count3+d^2
        fi:
    end do:
    count1-count3;
end proc: # Ridouane Oudra, Feb 21 2023
# second Maple program:
a:= n-> add(`if`(d::odd, d^2*(-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Ridouane Oudra, Feb 21 2023

Mathematica

QP = QPochhammer; s = (1-QP[q]^4*(QP[q^2]^6/QP[q^4]^4))/(4*q) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^2]^4) / 4, {q, 0, n}]; (* Michael Somos, Jun 25 2019 *)
f[p_, e_] := If[Mod[p, 4] == 1, ((p^2)^(e+1)-1)/(p^2-1), ((-p^2)^(e+1)-1)/(-p^2-1)]; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d)))} /* Michael Somos, Aug 09 2006 */
(Haskell)
a002173 n = a050450 n - a050453 n  -- Reinhard Zumkeller, Jun 17 2013

Crossrefs

Equals A050450(n) - A050453(n).

A120030(n) = -4*a(n), if n>0.

Cf. A056594.

Sequence in context: A019432 A211796 A138505 * A050458 A358877 A125166

Adjacent sequences: A002170 A002171 A002172 * A002174 A002175 A002176

Keyword

sign,easy,mult,look

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved