A002171 -id:A002171 - cp's OEIS frontend

A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144

Offset: 1

N. J. A. Sloane

Multiplicative: If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).
a(n) is the number of sublattices of index n in a generic 2-dimensional lattice. - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001 [In the language of group theory, a(n) is the number of index-n subgroups of Z x Z. - Jianing Song, Nov 05 2022]
The sublattices of index n are in one-to-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} d = sigma(n), which is a(n). A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * Product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]
Sum of number of common divisors of n and m, where m runs from 1 to n. - Naohiro Nomoto, Jan 10 2004
a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).
Let s(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + a(n-15) - a(n-22) - a(n-26) + ..., then a(n) = s(n) if n is not pentagonal, i.e., n != (3 j^2 +- j)/2 (cf. A001318), and a(n) is instead s(n) - ((-1)^j)*n if n is pentagonal. - Gary W. Adamson, Oct 05 2008 [corrected Apr 27 2012 by William J. Keith based on Ewell and by Andrey Zabolotskiy, Apr 08 2022]
Write n as 2^k * d, where d is odd. Then a(n) is odd if and only if d is a square. - Jon Perry, Nov 08 2012
Also total number of parts in the partitions of n into equal parts. - Omar E. Pol, Jan 16 2013
Note that sigma(3^4) = 11^2. On the other hand, Kanold (1947) shows that the equation sigma(q^(p-1)) = b^p has no solutions b > 2, q prime, p odd prime. - N. J. A. Sloane, Dec 21 2013, based on postings to the Number Theory Mailing List by Vladimir Letsko and Luis H. Gallardo
Limit_{m->infinity} (Sum_{n=1..prime(m)} a(n)) / prime(m)^2 = zeta(2)/2 = Pi^2/12 (A072691). See more at A244583. - Richard R. Forberg, Jan 04 2015
a(n) + A000005(n) is an odd number iff n = 2m^2, m>=1. - Richard R. Forberg, Jan 15 2015
a(n) = a(n+1) for n = 14, 206, 957, 1334, 1364 (A002961). - Zak Seidov, May 03 2016
Equivalent to the Riemann hypothesis: a(n) < H(n) + exp(H(n))*log(H(n)), for all n>1, where H(n) is the n-th harmonic number (Jeffrey Lagarias). See A057641 for more details. - Ilya Gutkovskiy, Jul 05 2016
a(n) is the total number of even parts in the partitions of 2*n into equal parts. More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 18 2019
From Jianing Song, Nov 05 2022: (Start)
a(n) is also the number of order-n subgroups of C_n X C_n, where C_n is the cyclic group of order n. Proof: by the correspondence theorem in the group theory, there is a one-to-one correspondence between the order-n subgroups of C_n X C_n = (Z x Z)/(nZ x nZ) and the index-n subgroups of Z x Z containing nZ x nZ. But an index-n normal subgroup of a (multiplicative) group G contains {g^n : n in G} automatically. The desired result follows from the comment from Naohiro Nomoto above.
The number of subgroups of C_n X C_n that are isomorphic to C_n is A001615(n). (End)

			For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.
Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 141, 166.
H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
Kanold, Hans Joachim, Kreisteilungspolynome und ungerade vollkommene Zahlen. (German), Ber. Math.-Tagung Tübingen 1946, (1947). pp. 84-87.
M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.
A. Lubotzky, Counting subgroups of finite index, Proceedings of the St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture Notes Series no. 212 Cambridge University Press 1995.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.1, page 77.
G. Pólya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 91, 395.
Robert M. Young, Excursions in Calculus, The Mathematical Association of America, 1992 p. 361.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane)
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B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
M. Baake and U. Grimm, Quasicrystalline combinatorics
Henry Bottomley, Illustration of initial terms
C. K. Caldwell, The Prime Glossary, sigma function
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012. - _N. J. A. Sloane_, Dec 25 2012
R. L. Duncan, Note on the Divisors of a Number, The American Mathematical Monthly, 68(4), 356-359, (1961).
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal (2004), Vol. 14.1, page 243.
L. Euler, Discovery of a most extraordinary law of numbers, relating to the sum of their divisors
L. Euler, Observatio de summis divisorum
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009.
J. A. Ewell, Recurrences for the sum of divisors, Proc. Amer. Math. Soc. 64 (2) 1977.
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
Johan Gielis and Ilia Tavkhelidze, The general case of cutting of GML surfaces and bodies, arXiv:1904.01414 [math.GM], 2019.
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]
M. J. Grady, A group theoretic approach to a famous partition formula, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
Douglas E. Iannucci, On sums of the small divisors of a natural number, arXiv:1910.11835 [math.NT], 2019.
Antti Karttunen, Scheme program for computing this sequence.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math.CO/0503436, 2005.
K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
Walter Nissen, Abundancy : Some Resources
P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - _N. J. A. Sloane_, Feb 23 2009
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 3.
Eric Weisstein's World of Mathematics, Divisor Function
Wikipedia, Divisor function
Index entries for sequences related to sublattices
Index entries for sequences related to sigma(n)
Index entries for "core" sequences

See A034885, A002093 for records. Bisections give A008438, A062731. Values taken are listed in A007609. A054973 is an inverse function.
For partial sums see A024916.
Row sums of A127093.
sigma_i (i=0..25): A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972, A281959.
Cf. A144736, A158951, A158902, A174740, A147843, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices), A039653, A088580, A074400, A083728, A006352, A002659, A083238, A000593, A050449, A050452, A051731, A027748, A124010, A069192, A057641, A001318.
Cf. A009194, A082062 (gcd(a(n),n) and its largest prime factor), A179931, A192795 (gcd(a(n),A001157(n)) and largest prime factor).
Cf. also A034448 (sum of unitary divisors).
Cf. A007955 (products of divisors).
Cf. A144613, A144614, A144615, A146076.
A001227, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
Cf. A001221, A054533.

A000203:=List([1..10^2],n->Sigma(n)); # Muniru A Asiru, Oct 01 2017

a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, May 07 2012

Magma
```
[SumOfDivisors(n): n in [1..70]];
```

[DivisorSigma(1,n): n in [1..70]]; // Bruno Berselli, Sep 09 2015

with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);

Table[ DivisorSigma[1, n], {n, 100}]
a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* Michael Somos, Apr 25 2013 *)

makelist(divsum(n),n,1,1000); /* Emanuele Munarini, Mar 26 2011 */

numlib::sigma(n)$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI
```
{a(n) = if( n<1, 0, sigma(n))};
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])};

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k)^2, x * O(x^n)), n))}; /* Michael Somos, Jan 29 2005 */

max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)); a(n) = polcoeff(ser,n)*n \\ Gottfried Helms, Aug 10 2009

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jan 03 2021

from math import prod
from sympy import factorint
def a(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024
(APL, Dyalog dialect) A000203 ← +/{ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð} ⍝ Antti Karttunen, Feb 20 2024

[sigma(n, 1) for n in range(1, 71)]  # Zerinvary Lajos, Jun 04 2009

(definec (A000203 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (- (expt p (+ 1 e)) 1) (- p 1)) (A000203 (A028234 n)))))) ;; Uses macro definec from http://oeis.org/wiki/Memoization#Scheme - Antti Karttunen, Nov 25 2017

(define (A000203 n) (let ((r (sqrt n))) (let loop ((i (inexact->exact (floor r))) (s (if (integer? r) (- r) 0))) (cond ((zero? i) s) ((zero? (modulo n i)) (loop (- i 1) (+ s i (/ n i)))) (else (loop (- i 1) s)))))) ;; (Stand-alone program) - Antti Karttunen, Feb 20 2024

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson, Aug 01 2001
For the following bounds and many others, see Mitrinovic et al. - N. J. A. Sloane, Oct 02 2017
If n is composite, a(n) > n + sqrt(n).
a(n) < n*sqrt(n) for all n.
a(n) < (6/Pi^2)*n^(3/2) for n > 12.
G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = Product_{n>=1} (1-x^n). - Joerg Arndt, Mar 14 2010
L.g.f.: -log(Product_{j>=1} (1-x^j)) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.
a(n) is odd iff n is a square or twice a square. - Robert G. Wilson v, Oct 03 2001
a(n) = a(n*prime(n)) - prime(n)*a(n). - Labos Elemer, Aug 14 2003 (Clarified by Omar E. Pol, Apr 27 2016)
a(n) = n*A000041(n) - Sum_{i=1..n-1} a(i)*A000041(n-i). - Jon Perry, Sep 11 2003
a(n) = -A010815(n)*n - Sum_{k=1..n-1} A010815(k)*a(n-k). - Reinhard Zumkeller, Nov 30 2003
a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - Reinhard Zumkeller, Nov 17 2004
Recurrence: n^2*(n-1)*a(n) = 12*Sum_{k=1..n-1} (5*k*(n-k) - n^2)*a(k)*a(n-k), if n>1. - Dominique Giard (dominique.giard(AT)gmail.com), Jan 11 2005
G.f.: Sum_{k>0} k * x^k / (1 - x^k) = Sum_{k>0} x^k / (1 - x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - Michael Somos, Apr 05 2003. See the Hardy-Wright reference, p. 312. first equation, and p. 250, Theorem 290. - Wolfdieter Lang, Dec 09 2016
For odd n, a(n) = A000593(n). For even n, a(n) = A000593(n) + A074400(n/2). - Jonathan Vos Post, Mar 26 2006
Equals the inverse Moebius transform of the natural numbers. Equals row sums of A127093. - Gary W. Adamson, May 20 2007
A127093 * [1/1, 1/2, 1/3, ...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, ...]. Row sums of triangle A135539. - Gary W. Adamson, Oct 31 2007
a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
a(n) = n*Sum_{k=1..n} A060642(n,k)/k*(-1)^(k+1). - Vladimir Kruchinin, Aug 10 2010
Dirichlet convolution of A037213 and A034448. - R. J. Mathar, Apr 13 2011
G.f.: A(x) = x/(1-x)*(1 - 2*x*(1-x)/(G(0) - 2*x^2 + 2*x)); G(k) = -2*x - 1 - (1+x)*k + (2*k+3)*(x^(k+2)) - x*(k+1)*(k+3)*((-1 + (x^(k+2)))^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2011
a(n) = A001065(n) + n. - Mats Granvik, May 20 2012
a(n) = A006128(n) - A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A196020(n,k). - conjectured by Omar E. Pol, Feb 02 2013, and proved by Max Alekseyev, Nov 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A000330(k)*A000716(n-A000217(k)). - Mircea Merca, Mar 05 2014
a(n) = A240698(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014
a(n) = Sum_{d^2|n} A001615(n/d^2) = Sum_{d^3|n} A254981(n/d^3). - Álvar Ibeas, Mar 06 2015
a(3*n) = A144613(n). a(3*n + 1) = A144614(n). a(3*n + 2) = A144615(n). - Michael Somos, Jul 19 2015
a(n) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - Michel Lagneau, Oct 14 2015
a(n) = A000593(n) + A146076(n). - Omar E. Pol, Apr 05 2016
a(n) = A065475(n) + A048050(n). - Omar E. Pol, Nov 28 2016
a(n) = (Pi^2*n/6)*Sum_{q>=1} c_q(n)/q^2, with the Ramanujan sums c_q(n) given in A054533 as a c_n(k) table. See the Hardy reference, p. 141, or Hardy-Wright, Theorem 293, p. 251. - Wolfdieter Lang, Jan 06 2017
G.f. also (1 - E_2(q))/24, with the g.f. E_2 of A006352. See e.g., Hardy, p. 166, eq. (10.5.5). - Wolfdieter Lang, Jan 31 2017
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(n) + A162296(n).
a(n) = A092261(n) * A295294(n). [This can be further expanded, see comment in A291750.] (End)
a(n) = A000593(n) * A038712(n). - Ivan N. Ianakiev and Omar E. Pol, Nov 26 2017
a(n) = Sum_{q=1..n} c_q(n) * floor(n/q), where c_q(n) is the Ramanujan's sum function given in A054533. - Daniel Suteu, Jun 14 2018
a(n) = Sum_{k=1..n} gcd(n, k) / phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 21 2018
a(n) = (2^(1 + (A000005(n) - A001227(n))/(A000005(n) - A183063(n))) - 1)*A000593(n) = (2^(1 + (A183063(n)/A001227(n))) - 1)*A000593(n). - Omar E. Pol, Nov 03 2018
a(n) = Sum_{i=1..n} tau(gcd(n, i)). - Ridouane Oudra, Oct 15 2019
From Peter Bala, Jan 19 2021: (Start)
G.f.: A(x) = Sum_{n >= 1} x^(n^2)*(x^n + n*(1 - x^(2*n)))/(1 - x^n)^2 - differentiate equation 5 in Arndt w.r.t. x, and set x = 1.
A(x) = F(x) + G(x), where F(x) is the g.f. of A079667 and G(x) is the g.f. of A117004. (End)
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
With the convention that a(n) = 0 for n <= 0 we have the recurrence a(n) = t(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where t(n) = (-1)^(m+1)*(2*m+1)*n/3 if n = m*(m + 1)/2, with m positive, is a triangular number else t(n) = 0. For example, n = 10 = (4*5)/2 is a triangular number, t(10) = -30, and so a(10) = -30 + 3*a(9) - 5*a(7) + 7*a(4) = -30 + 39 - 40 + 49 = 18. - Peter Bala, Apr 06 2022
Recurrence: a(p^x) = p*a(p^(x-1)) + 1, if p is prime and for any integer x. E.g., a(5^3) = 5*a(5^2) + 1 = 5*31 + 1 = 156. - Jules Beauchamp, Nov 11 2022
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = A319462. - Vaclav Kotesovec, May 07 2023
a(n) < (7n*A001221(n) + 10*n)/6 [Duncan, 1961] (see Duncan and Tattersall). - Stefano Spezia, Jul 13 2025

A002654 Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2, 0, 0, 2, 0

Offset: 1

N. J. A. Sloane

Glaisher calls this E(n) or E_0(n). - N. J. A. Sloane, Nov 24 2018
Number of sublattices of Z X Z of index n that are similar to Z X Z; number of (principal) ideals of Z[i] of norm n.
a(n) is also one fourth of the number of integer solutions of n = x^2 + y^2 (order and signs matter, and 0 (without signs) is allowed). a(n) = N(n)/4, with N(n) from p. 147 of the Niven-Zuckermann reference. See also Theorem 5.12, p. 150, which defines a (strongly) multiplicative function h(n) which coincides with A056594(n-1), n >= 1, and N(n)/4 = sum(h(d), d divides n). - Wolfdieter Lang, Apr 19 2013
a(2+8*N) = A008441(N) gives the number of ways of writing N as the sum of 2 (nonnegative) triangular numbers for N >= 0. - Wolfdieter Lang, Jan 12 2017
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -4. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			4 = 2^2, so a(4) = 1; 5 = 1^2 + 2^2 = 2^2 + 1^2, so a(5) = 2.
x + x^2 + x^4 + 2*x^5 + x^8 + x^9 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + x^18 + ...
2 = (+1)^2 + (+1)^2 = (+1)^2 + (-1)^2  = (-1)^2 + (+1)^2 = (-1)^2 + (-1)^2. Hence there are 4 integer solutions, called N(2) in the Niven-Zuckerman reference, and a(2) = N(2)/4 = 1.  4 = 0^1 + (+2)^2 = (+2)^2 + 0^2 = 0^2 + (-2)^2 = (-2)^2 + 0^2. Hence N(4) = 4 and a(4) = N(4)/4 = 1. N(5) = 8, a(5) = 2. - _Wolfdieter Lang_, Apr 19 2013

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194.
George 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(17). MR0121327 (22 #12066)
Emil Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley, 1980, pp. 147 and 150.
Günter Scheja and Uwe Storch, Lehrbuch der Algebra, Tuebner, 1988, p. 251.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 89.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 340.

T. D. Noe, Table of n, a(n) for n = 1..10000
Michael Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., The Mathematics of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44; arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Uwe Grimm, Quasicrystalline combinatorics, 2002.
Shai Covo, Problem 3586, Crux Mathematicorum, Vol. 36, No. 7 (2010), pp. 461 and 463; Solution to Problem 3586 by the proposer, ibid., Vol. 37, No. 7 (2011), pp. 477-479.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, Vol. 20 (1884), pp. 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, Vol. 20 (1884), pp. 97-167. [Annotated scanned copy]
J. W. L. Glaisher, On the function which denotes the difference between the number of (4m+1)-divisors and the number of (4m+3)-divisors of a number, Proc. London Math. Soc., Vol. 15 (1884), pp. 104-122.
J. W. L. Glaisher, On the function which denotes the difference between the number of (4m+1)-divisors and the number of (4m+3)-divisors of a number, Proc. London Math. Soc., Vol. 15 (1884), pp. 104-122. [Annotated scanned copy of pages 104-107 only]
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., Vol. 38 (1907), pp. 1-62 (see p. 4 and p. 8).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 5.
Vaclav Kotesovec, Graph - the asymptotic ratio of Sum_{k=1..n} a(k)^2
Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., Vol. 6 (2002), pp. 7-149.
PeakMath, L-functions and the Langlands program (RH Saga S1E2), YouTube video (2024).
Srinivasa Ramanujan, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, section (K).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009
Index entries for sequences related to sums of squares.
Index entries for sequences related to sublattices.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000161, A001481, A003881.
Equals 1/4 of A004018. Partial sums give A014200.
Cf. A002175, A008441, A121444, A122856, A122865, A022544, A143574, A000265, A027748, A124010, A025426 (two squares, order does not matter), A120630 (Dirichlet inverse), A101455 (Mobius transform), A000089, A241011.
If one simply reads the table in Glaisher, PLMS 1884, which omits the zero entries, one gets A213408.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

a002654 n = product $ zipWith f (a027748_row m) (a124010_row m) where
   f p e | p `mod` 4 == 1 = e + 1
         | otherwise      = (e + 1) `mod` 2
   m = a000265 n
-- Reinhard Zumkeller, Mar 18 2013

with(numtheory):
A002654 := 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+1
        elif d mod 4 = 3 then
            count3 := count3+1
        fi:
    end do:
    count1-count3;
end proc:
# second Maple program:
a:= n-> add(`if`(d::odd, (-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 04 2020

a[n_] := Count[Divisors[n], d_ /; Mod[d, 4] == 1] - Count[Divisors[n], d_ /; Mod[d, 4] == 3]; a/@Range[105] (* Jean-François Alcover, Apr 06 2011, after R. J. Mathar *)
QP = QPochhammer; CoefficientList[(1/q)*(QP[q^2]^10/(QP[q]*QP[q^4])^4-1)/4 + O[q]^100, q] (* Jean-François Alcover, Nov 24 2015 *)
f[2, e_] := 1; f[p_, e_] := If[Mod[p, 4] == 1, e + 1, Mod[e + 1, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)
Rest[CoefficientList[Series[EllipticTheta[3, 0, q]^2/4, {q, 0, 100}], q]] (* Vaclav Kotesovec, Mar 10 2023 *)

direuler(p=2,101,1/(1-X)/(1-kronecker(-4,p)*X))

{a(n) = polcoeff( sum(k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}

{a(n) = sumdiv( n, d, (d%4==1) - (d%4==3))}

{a(n) = local(A); A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / (eta(x + A) * eta(x^4 + A))^4 / 4, n)} \\ Michael Somos, Jun 03 2005

a(n)=my(f=factor(n>>valuation(n,2))); prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, (f[i,2]+1)%2)) \\ Charles R Greathouse IV, Sep 09 2014

my(B=bnfinit(x^2+1)); vector(100,n,#bnfisintnorm(B,n)) \\ Joerg Arndt, Jun 01 2024

from math import prod
from sympy import factorint
def A002654(n): return prod(1 if p == 2 else (e+1 if p % 4 == 1 else (e+1) % 2) for p, e in factorint(n).items()) # Chai Wah Wu, May 09 2022

Dirichlet series: (1-2^(-s))^(-1)*Product (1-p^(-s))^(-2) (p=1 mod 4) * Product (1-p^(-2s))^(-1) (p=3 mod 4) = Dedekind zeta-function of Z[ i ].
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -16.
If n=2^k*u*v, where u is product of primes 4m+1, v is product of primes 4m+3, then a(n)=0 unless v is a square, in which case a(n) = number of divisors of u (Jacobi).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p == 1 (mod 4); (e+1) mod 2 if p == 3 (mod 4). - David W. Wilson, Sep 01 2001
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * (4*w + 1). - Michael Somos, Jul 19 2004
G.f.: Sum_{n>=1} ((-1)^floor(n/2)*x^((n^2+n)/2)/(1+(-x)^n)). - Vladeta Jovovic, Sep 15 2004
Expansion of (eta(q^2)^10 / (eta(q) * eta(q^4))^4 - 1)/4 in powers of q.
G.f.: Sum_{k>0} x^k / (1 + x^(2*k)) = Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 - x^(2*k - 1)). - Michael Somos, Aug 17 2005
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(2*n) = a(n). - Michael Somos, Nov 01 2006
a(4*n + 1) = A008441(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n). 4 * a(n) = A004018(n) unless n=0.
a(n) = Sum_{k=1..n} A010052(k)*A010052(n-k). a(A022544(n)) = 0; a(A001481(n)) > 0.
- Reinhard Zumkeller, Sep 27 2008
a(n) = A001826(n) - A001842(n). - R. J. Mathar, Mar 23 2011
a(n) = Sum_{d|n} A056594(d-1), n >= 1. See the above comment on A056594(d-1) = h(d) of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
Dirichlet g.f.: zeta(s)*beta(s) = zeta(s)*L(chi_2(4),s). - Ralf Stephan, Mar 27 2015
G.f.: (theta_3(x)^2 - 1)/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
a(n) = Sum_{ m: m^2|n } A000089(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = A053866(n) + 2 * A025441(n). - Andrey Zabolotskiy, Apr 23 2019
a(n) = Im(Sum_{d|n} i^d). - Ridouane Oudra, Feb 02 2020
a(n) = Sum_{d|n} sin((1/2)*d*Pi). - Ridouane Oudra, Jan 22 2021
Sum_{n>=1} (-1)^n*a(n)/n = Pi*log(2)/4 (Covo, 2010). - Amiram Eldar, Apr 07 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 = 0.785398... (A003881). - Amiram Eldar, Oct 11 2022
From Vaclav Kotesovec, Mar 10 2023: (Start)
Sum_{k=1..n} a(k)^2 ~ n * (log(n) + C) / 4, where C = A241011 =
4*gamma - 1 + log(2)/3 - 2*log(Pi) + 8*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.01662154573340811526279685971511542645018417752364748061...
The constant C, published by Ramanujan (1916, formula (22)), 4*gamma - 1 + log(2)/3 - log(Pi) + 4*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.3482276258576... is wrong! (End)

A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.

Original entry on oeis.org

1, -6, 9, 10, -30, 0, 11, 42, 0, -70, 18, -54, 49, 90, 0, -22, -60, 0, -110, 0, 81, 180, -78, 0, 130, -198, 0, -182, -30, 90, 121, 84, 0, 0, 210, 0, -252, -102, -270, 170, 0, 0, -69, 330, 0, -38, 420, 0, -190, -390, 0, -108, 0, 0, 0, -300, 99, 442, 210, 0, 418, -294, 0, 0, -510, 378, -540, 138, 0

Offset: 0

N. J. A. Sloane

This is Glaisher's function lambda(m). It appears to be defined only for odd m, and lambda(4t-1) = 0 (t >= 1), lambda(4t+1) = a(t) (t >= 0). - N. J. A. Sloane, Nov 25 2018
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 36 of the 74 eta-quotients listed in Table I of Martin (1996).
Dickson, v.2, p. 295 briefly states a result of Glaisher, 1883, pp 212-215. This result is that a(n) is the sum over all solutions of 16*n + 4 = x^2 + y^2 + z^2 + w^2 in nonnegative odd integers of chi(x) and is also the sum over all solutions of 8*n + 2 = x^2 + y^2 in nonnegative odd integers of chi(x) * chi(y) where chi(x) = x if x == 1 (mod 4) and -x if x == 3 (mod 4). [Michael Somos, Jun 18 2012]
Denoted by g_3(q) in Cynk and Hulek on page 8 as the unique weight 3 Hecke eigenform of level 16 with complex multiplication by i. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - Michael Somos, Aug 24 2012

			G.f. = 1 - 6*x + 9*x^2 + 10*x^3 - 30*x^4 + 11*x^6 + 42*x^7 - 70*x^9 + 18*x^10 + ...
G.f. = q - 6*q^5 + 9*q^9 + 10*q^13 - 30*q^17 + 11*q^25 + 42*q^29 - 70*q^37 + ...

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 295, and vol. 3, p. 134.
J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See page 340.
J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega(m), Quart. J. Math, 37 (1906), 36-48.
Morris Newman, A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
S. Cooper, M. D. Hirschhorn and R. Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
J. W. L. Glaisher, Note on the Compositions of a Number as a Sum of Two and Four Uneven Squares, Quarterly Journal of Pure and Applied Mathematics, 19 (1883), 212-215.
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. 5).
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
S. Milne and V. Leininger, Some new infinite families of eta function identities, Methods and Applications of Analysis 6 (1999), 225--248.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
Index entries for sequences mentioned by Glaisher

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

A := Basis( ModularForms( Gamma1(16), 3), 274); A[2] - 6*A[6] + 9*A[10] + 10*A[14] - 30*A[18]; /* Michael Somos, May 17 2015 */

A := Basis( CuspForms( Gamma1(16), 3), 274); A[1] - 6*A[5]; /* Michael Somos, Jan 09 2017 */

a[ n_] := SeriesCoefficient[ 1/16 EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q]^4 EllipticTheta[ 3, 0, q], {q, 0, 4 n + 1}]; (* Michael Somos, Jun 18 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[ 16 n + 4]}, SeriesCoefficient[ Sum[ Mod[k, 2] q^k^2, {k, m}]^3 Sum[ KroneckerSymbol[ -4, k] k q^k^2, {k, m}], {q, 0, 16 n + 4}]]]; (* Michael Somos, Jun 12 2012 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^3 / (4 q^(1/2)), {q, 0, 2 n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6, {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/4) EllipticThetaPrime[ 1, -Pi/4, q] EllipticTheta[ 1, -Pi/4, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (-1/16) EllipticThetaPrime[ 1, 0, q] EllipticTheta[ 1, -Pi/2, q]^3, {q, 0, 4 n + 1}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%4==3, if( e%2, 0, p^e), forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0=1; a1 = y = 2*(x^2 - y^2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 21 2006 */

{a(n)=local(tn=(sqrtint(8*n+1)+1)\2);polcoeff(sum(m=0,tn,(1+2*m)^2*x^(m^2+m)+x*O(x^n)) + 2*sum(m=0,tn,sum(k=1,tn,(1+4*(m^2+m-k^2))*x^(m^2+m+k^2)+x*O(x^n))),n)} /* Paul D. Hanna, Mar 15 2010 */

Expansion of q^(-1/4)/16 * theta_2(q)^4 * theta_3(q) * theta_4(q) in powers of q. - [Dickson, v. 3, p. 134] from Stieltjes footnote 160. Michael Somos, Jun 18 2012
Expansion of q^(-1/2) / 4 * k * k' * (K / (pi/2))^3 in powers of q^2 where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
G.f.: Product_{k>0}(1 - x^k)^6.
Given g.f. A(x), then A(q^4) = f(-q^4)^6 = phi(q) * phi(-q) * psi(q^2)^4 where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 23 2006
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^2 if p == 1 (mod 4) and b(p) = 2 * (x^2 - y^2) where p = x^2 + y^2 and y is even. - Michael Somos, Aug 23 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 24 2012
G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (1/2) * Sum_{u,v in Z} (u*u - 4*v*v) * x^(u*u + 4*v*v). - Michael Somos, Jun 14 2007
G.f.: eta(x)^6 = Sum_{n>=0} (1+2n)^2*x^(n^2+n) + 2*Sum_{n>=0,k>=1} (1 + 4(n^2+n-k^2))*x^(n^2+n+k^2) - from the Milne and Leininger reference. [Paul D. Hanna, Mar 15 2010]
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-6*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M be a positive integer whose prime factors are all congruent to 3 (mod 4) - see A004614. Then a( M^2*n + (M^2 - 1)/4 ) = M^2*a(n). See Cooper et al., equation 5. - Peter Bala, Dec 01 2020
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = ((x+y*i)^(2*e+2) - (x-y*i)^(2*e+2))/((x+y*i)^2 - (x-y*i)^2) if p == 1 (mod 4) where p = x^2 + y^2 and x is odd. - Jianing Song, Mar 19 2022

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

Original entry on oeis.org

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

N. J. A. Sloane

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

			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

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).

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.

Cf. A050450, A050453, A120030.

Cf. A056594.

a002173 n = a050450 n - a050453 n  -- Reinhard Zumkeller, Jun 17 2013

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

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 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-4, d)))} /* Michael Somos, Aug 09 2006 */

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

a(n) = A050450(n) - A050453(n).
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

More terms from David W. Wilson

A030211 Expansion of q^(-1/2) * (eta(q) * eta(q^2))^4 in powers of q.

Original entry on oeis.org

1, -4, -2, 24, -11, -44, 22, 8, 50, 44, -96, -56, -121, 152, 198, -160, 176, -48, -162, -88, -198, 52, 22, 528, 233, -200, -242, 88, -176, -668, 550, -264, -44, 188, 224, 728, 154, 484, -1056, -656, -311, 236, -100, -792, 714, 528, 640, -88, -478, 484, 1566, -968, 192, -780, -1994, 648, -942

Offset: 0

N. J. A. Sloane

This is Glaisher's P(n). - N. J. A. Sloane, Nov 24 2018

Number 16 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 4*x - 2*x^2 + 24*x^3 - 11*x^4 - 44*x^5 + 22*x^6 + 8*x^7 + 50*x^8 + ...
G.f. = q - 4*q^3 - 2*q^5 + 24*q^7 - 11*q^9 - 44*q^11 + 22*q^13 + 8*q^15 + ...

J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See p. 340.
J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega(m). Quart. J. Math, 37 (1906), 36-48.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
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. 5).
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Eric Weisstein's World of Mathematics, Apéry Number.
Index entries for sequences mentioned by Glaisher

Cf. A002171, A134461 (the same except for signs).

Basis( CuspForms( Gamma0(8), 4), 115) [1]; /* Michael Somos, May 27 2014 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^4, {x, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = if( n<0, 0, polcoeff( (eta(x + x * O(x^n)) * eta(x^2 + x * O(x^n)))^4, n))}; /* Michael Somos, Apr 14 2004 */

q='q+O('q^99); Vec((eta(q)*eta(q^2))^4) \\ Altug Alkan, Sep 19 2018

CuspForms( Gamma0(8), 4, prec=115).0; # Michael Somos, May 28 2013

G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(2*k)))^4.
Euler transform of period 2 sequence [ -4, -8, ...]. - Michael Somos, Apr 14 2004
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (81*u6*u3 + u1*u2) * (u2*u3 + u1*u6) + 30 * u1*u2*u3*u6 - 256 * u2^2*u6^2 - 5 * u2^2*u3^2 - 5 * u1^2*u6^2 - u1^2*u3^2. - Michael Somos, Mar 08 2006
Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = 8 * A0*A2 * (A0^2 + A2^2) + (A1^2 - A3^2) * (A0^2 - A2^2). - Michael Somos, Mar 08 2006
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2. - Michael Somos, Mar 08 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 64 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, May 28 2013
a(n) = (-1)^n * A134461(n). Convolution square of A002171.
G.f.: exp(4*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A002172 Glaisher's chi numbers chi(p) for p a prime of the form 4m+1.

Original entry on oeis.org

-2, 6, 2, -10, -2, 10, 14, -10, -6, 10, 18, -2, 6, -14, -22, 14, 22, -26, -18, -14, -2, 30, 26, -30, 2, -26, -18, 10, -34, 26, 22, 18, -10, 34, 14, -34, 38, 2, -6, 30, 34, -14, 42, 38, -10, -22, -42, 38, 26, 2, -46, 10, -34, -38, 50, -26, -50, -46, -2, -10, 30, 54, -18, -38, 50, -34, 22, 10, -50, 54, 46, 58, -58, 50

Offset: 1

N. J. A. Sloane

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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]
Index entries for sequences related to Glaisher's numbers

Cf. A002171.

pp = Select[ Prime[ Range[200]], Mod[#, 4] == 1 & ]; (-Sum[ JacobiSymbol[x^3 - x, #], {x, 0, # - 1}] & ) /@ pp (* Jean-François Alcover, Oct 07 2011, after Michael Somos *)

{a(n)= local(m, c); if(n<1, 0, c=0; m=0; while(cMichael Somos, Sep 19 2006 */

A215472 Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -14, 81, -238, 322, 0, -429, 82, 0, 2162, -3038, -1134, 2401, 2482, 0, -6958, 3332, 0, 1442, 0, 6561, -4508, -9758, 0, -1918, 18802, 0, 9362, -24638, -19278, 14641, 14756, 0, 0, 6562, 0, -1148, -33998, 26082, -20398, 0, 0, 28083, 49042, 0, -64078, -30268

Offset: 0

Michael Somos, Aug 12 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

This is a member of an infinite family of integer weight level 8 modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

			1 - 14*x + 81*x^2 - 238*x^3 + 322*x^4 - 429*x^6 + 82*x^7 + 2162*x^9 + ...
q - 14*q^5 + 81*q^9 - 238*q^13 + 322*q^17 - 429*q^25 + 82*q^29 + 2162*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shobhit, How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A030212, A209942, A215601.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^14 / QPochhammer[ x^2]^4, {x, 0, n}] (* Michael Somos, Sep 05 2013 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^7 / eta(x^2 + A)^2 )^2, n))}

Expansion of q^(-1/4) * eta(q)^14 / eta(q^2)^4 in powers of q.
Expansion of q^(-1/4) * ( eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 + 4 * eta(q^2)^4 * eta(q^4)^2 * eta(q^8)^4 ) in powers of q. - Michael Somos, Sep 05 2013
Euler transform of period 2 sequence [ -14, -10, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 128 (t/i)^5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030212.
a(n) = (-1)^n * A209942(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n).
a(n) = A030212(4*n + 1). - Michael Somos, Sep 05 2013

A215601 Expansion of phi(-x)^2 * f(-x)^6 + 32 * x * psi(-x)^2 * f(-x^4)^6 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 22, -27, -18, -94, 0, 359, -130, 0, 214, -230, -594, -343, 518, 0, 830, -396, 0, 1098, 0, 729, -2068, -1670, 0, 594, 598, 0, -1746, 2002, 486, -1331, 5148, 0, 0, -1606, 0, -2860, -3514, 2538, 286, 0, 0, -1873, -4082, 0, 3942, 4708, 0, 5362, 1174, 0, -5060

Offset: 0

Michael Somos, Aug 16 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by g_4(q) in Cynk and Hulek on page 8 as the unique weight 4 Hecke eigenform of level 32 with complex multiplication by i. - Michael Somos, Aug 24 2012
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

			G.f. = 1 + 22*x - 27*x^2 - 18*x^3 - 94*x^4 + 359*x^6 - 130*x^7 + 214*x^9 - 230*x^10 + ..
G.f. = q + 22*q^5 - 27*q^9 - 18*q^13 - 94*q^17 + 359*q^25 - 130*q^29 + 214*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A215472, A215600.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2 + 32 x (QPochhammer[ x] QPochhammer[ x^4]^4 / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jan 11 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^5 / eta(x^2 + A) )^2 + 32 * x * ( eta(x + A) * eta(x^4 + A)^4 / eta(x^2 + A) )^2, n))};

{a(n) = local(A, p, e, x, y, a0, a1, w=3); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p%4==3, if( e%2, 0, (-p)^(w*e/2)), y=-sum( i=0, p-1, kronecker( i^3-i, p)); a0=2; a1=y; for( i=2, w, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for( i=2, e, x=y*a1 -p^w*a0; a0=a1; a1=x); a1)))))};

Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 + 32 * (eta(q) * eta(q^4)^4 / eta(q^2))^2 in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^10 (t/i)^4 f(t) where q = exp(2 Pi i t).
a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -27 * a(n). a(n) = A215600(2*n).

A279955 Expansion of chi(-x^4)^4 * f(-x^4)^2 * f(-x)^2 in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, -1, 2, -5, 14, 4, -12, 5, -40, 0, 26, 11, 68, -15, -30, -18, -106, 3, 50, -10, 182, 29, -104, 10, -270, 11, 130, 37, 360, -51, -164, -16, -506, -30, 266, -65, 686, 62, -320, 53, -898, 22, 414, 50, 1206, -61, -612, -52, -1560, -4, 696, -81, 1958, 120

Offset: 0

Michael Somos, Dec 23 2016

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 2*x - x^2 + 2*x^3 - 5*x^4 + 14*x^5 + 4*x^6 - 12*x^7 + 5*x^8 + ...
G.f. = q^-1 - 2*q^3 - q^7 + 2*q^11 - 5*q^15 + 14*q^19 + 4*q^23 - 12*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002171, A280339.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^4]^2 QPochhammer[ x^4, x^8]^4, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^6 / eta(x^8 + A)^4, n))};

Expansion of q * eta(q^4)^2 * eta(q^16)^6 / eta(q^32)^4 in powers of q^4.
Euler transform of period 8 sequence [ -2, -2, -2, -8, -2, -2, -2, -4, ...].
a(n) = (-1)^n * A280339(n).
a(3*n + 1) / a(1) == A002171(n) (mod 3). a(3^3*n + 7) / a(7) == A002171(n) (mod 3^2).

A138515 Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q.

Original entry on oeis.org

1, 2, -3, -6, 2, 0, -1, 10, 0, 2, 10, -6, -7, -14, 0, 10, -12, 0, -6, 0, 9, 4, 10, 0, 18, 2, 0, -6, -14, 18, -11, -12, 0, 0, -22, 0, 20, -14, -6, -22, 0, 0, 23, 26, 0, 18, 4, 0, -14, 2, 0, 20, 0, 0, 0, -12, 3, -30, 26, 0, -30, -14, 0, 0, 2, -30, -28, 26, 0, 18, 10, 0, -13, 34, 0, 0, 20, 0, 26, -22, 0, 6, 0, -6, 18, 0

Offset: 0

Michael Somos, Mar 22 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 58 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 16 2012
The weight 2 eta-quotient newform  eta^8(8*z) / (eta^2(4*z)*eta^2(16*z)) appears in Theorem 2 of the Martin and Ono link in the row with conductor 64 for the strong Weil curve y^2 = x^3 - 4*x. For N(p), the number of solutions modulo primes for this elliptic curve and for y^2 = x^3 + x, see A095978. The non-vanishing p-defects p - N(p) for these two curves are given in A267859.  - Wolfdieter Lang, May 26 2016

			G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 - x^6 + 10*x^7 + 2*x^9 + 10*x^10 - 6*x^11 + ...
G.f. for {b(n)} = q + 2*q^5 - 3*q^9 - 6*q^13 + 2*q^17 - q^25 + 10*q^29 + 2*q^37 + 10*q^41 - 6*q^45 - 7*q^49 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
LMFDB, Elliptic Curve 64.a4 (Cremona label 64a4)
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002171, A095978, A138514, A267859, A280339.

A := Basis( CuspForms( Gamma0(64), 2), 342); A[1] + 2*A[3]; /* Michael Somos, May 15 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ -q])^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^4 / (QPochhammer[ q] QPochhammer[ q^4]))^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)

{a(n) = ellak( ellinit( [ 0, 0, 0, 1, 0], 1), 4*n + 1)};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)))^2, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%4==1, forstep( x=1, sqrtint(p), 2, if( issquare( p - x^2), y=x; break)); y = 2 * y * (2 - (y%4)); a0 = 1; a1 = y; for(i=2, e, x = y * a1 - p * a0; a0 = a1; a1 = x); a1, if( e%2==0, (-p)^(e / 2)))))};

Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x.
Expansion of f(q)^2 * f(-q^2)^2 = psi(-q)^2 * phi(q)^2 = chi(q)^2 * f(-q^2)^4 = psi(q)^2 * phi(-q^2)^2 = f(q)^4 / chi(q)^2 = f(q)^6 / phi(q)^2 = f(-q^2)^6 / psi(-q)^2 = phi(q)^4 / chi(q)^6 = chi(q)^6 * psi(-q)^4 = f(q)^3 * psi(-q) = f(-q^2)^3 * phi(q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [2, -6, 2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 4) with b(p) = 2 * x * (-1)^((x-1)/2) where p = x^2 + 4 * y^2.
G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)))^2.
a(n) = (-1)^n * A002171(n). a(9*n + 2) = -3 * a(n), a(9*n + 5) = a(9*n + 8) = 0. Convolution square of A138514.
G.f. for{b(n)}:
  eta^8(8*z)/(eta^2(4*z)*eta^2(16*z)) with q = exp(2*Pi*i*z)), Im(z) > 0 (see a comment on the Martin-Ono link above). - Wolfdieter Lang, May 27 2016

cp's OEIS Frontend

A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

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A002654 Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).

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A000729 Expansion of Product_{k >= 1} (1 - x^k)^6.

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

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