A321546 -id:A321546 - cp's OEIS frontend

A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

1, -3, 10, -19, 26, -30, 50, -83, 91, -78, 122, -190, 170, -150, 260, -339, 290, -273, 362, -494, 500, -366, 530, -830, 651, -510, 820, -950, 842, -780, 962, -1363, 1220, -870, 1300, -1729, 1370, -1086, 1700, -2158, 1682, -1500, 1850, -2318, 2366, -1590, 2210, -3390, 2451, -1953, 2900, -3230, 2810, -2460, 3172

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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).
Index entries for sequences mentioned by Glaisher.

Apart from signs, same as A064027.
Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A002614 Glaisher's function theta(n) (18 squares version).

Original entry on oeis.org

0, -7, 128, -975, 4608, -16340, 48384, -124303, 281600, -583746, 1146240, -2125108, 3691008, -6151880, 10055424, -15914895, 24136704, -35748899, 52583040, -75877938, 105994240, -145580124, 200279808, -272040500, 359036928, -468767690, 615599360

Offset: 1

N. J. A. Sloane

sign

It would be nice to have a q-series that generates this sequence. Glaisher gives many formulas but they are difficult to follow.

J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Math. 38 (1907), 289-351 (see p. 349). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
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).

Sean A. Irvine, Table of n, a(n) for n = 1..500
Index entries for sequences mentioned by Glaisher

Cf. A002288, A321546.

a(n) = (A321546(n) - A002288(n)) / 17. - Sean A. Irvine, Mar 04 2019

Edited and signs added by N. J. A. Sloane, Nov 26 2018

More terms from Sean A. Irvine, Mar 04 2019

cp's OEIS Frontend

A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

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