A002173 a(n) = Sum_{d|n, d == 1 mod 4} d^2 - Sum_{d|n, d == 3 mod 4} d^2.
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
Examples
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
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.
Programs
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Haskell
a002173 n = a050450 n - a050453 n -- Reinhard Zumkeller, Jun 17 2013
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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
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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 *)
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PARI
{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d)))} /* Michael Somos, Aug 09 2006 */ -
Python
from math import prod from sympy import factorint def A002173(n): return prod(((m:=p**2*(0,1,0,-1)[p&3])**(e+1)-1)//(m-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024
Formula
A120030(n) = -4*a(n), if n>0.
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
G.f.: Sum_{n>=0} (4*n + 1)^2*x^(4*n + 1)/(1 - x^(4*n + 1)) - (4*n + 3)^2*x^(4*n + 3)/(1 - x^(4*n + 3)). - Miles Wilson, Oct 26 2024
Extensions
More terms from David W. Wilson
Comments