A098108 -id:A098108 - cp's OEIS frontend

A126811 Duplicate of A098108.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

dead

A010054 a(n) = 1 if n is a triangular number, otherwise 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

This is essentially the q-expansion of the Jacobi theta function theta_2(q). (In theta_2 one has to ignore the initial factor of 2*q^(1/4) and then replace q by q^(1/2). See also A005369.) - N. J. A. Sloane, Aug 03 2014
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's theta function f(a, b) = Sum_{n=-inf..inf} a^(n*(n+1)/2) * b^(n*(n-1)/2).
This sequence is the concatenation of the base-b digits in the sequence b^n, for any base b >= 2. - Davis Herring (herring(AT)lanl.gov), Nov 16 2004
Number of partitions of n into distinct parts such that the greatest part equals the number of all parts, see also A047993; a(n)=A117195(n,0) for n > 0; a(n) = 1-A117195(n,1) for n > 1. - Reinhard Zumkeller, Mar 03 2006
Triangle T(n,k), 0 <= k <= n, read by rows, given by A000007 DELTA A000004 where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Convolved with A000041 = A022567, the convolution square of A000009. - Gary W. Adamson, Jun 11 2009
A008441(n) = Sum_{k=0..n} a(k)*a(n-k). - Reinhard Zumkeller, Nov 03 2009
Polcoeff inverse with alternate signs = A006950: (1, 1, 1, 2, 3, 4, 5, 7, ...). - Gary W. Adamson, Mar 15 2010
This sequence is related to Ramanujan's two-variable theta functions because this sequence is also the characteristic function of generalized hexagonal numbers. - Omar E. Pol, Jun 08 2012
Number 3 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016
Number of partitions of n into consecutive parts that contain 1 as a part, n >= 1. - Omar E. Pol, Nov 27 2020
The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). The constant whose expansion in any base b >= 2 is this sequence is irrational (Bundschuh, 1984). - Amiram Eldar, Mar 23 2025

			G.f. = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 + x^45 + x^55 + x^66 + ...
G.f. for B(q) = q * A(q^8): q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + q^361 + ...
From _Philippe Deléham_, Jan 04 2008: (Start)
As a triangle this begins:
  1;
  1, 0;
  1, 0, 0;
  1, 0, 0, 0;
  1, 0, 0, 0, 0;
  1, 0, 0, 0, 0, 0;
  ...  (End)

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 103.
Michael D. Hirschhorn, The Power of q, Springer, 2017. See Psi, page 9.
Jules Tannery and Jules Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
Edmund T. Whittaker and George N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum, Vol. 17, No. 3 (2022), 129-141. See Conjecture 4.4, p. 137.
Peter Bundschuh, Generalization of a recent irrationality result of Mahler, Journal of Number Theory, Vol. 19, No. 2 (1984), pp. 248-253.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math. 274 (2004), no. 1-3, 9-24. See psi(q).
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See p. 2.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 2.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Michael D. Hirschhorn and James A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.
Ken Ono, Sinai Robins, and Patrick T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, Volume 50, Issue 1-2 (August 1995), pp. 73-94, Proposition 1; ResearchGate link; author's copy.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Michael Somos, A Multisection of q-Series, 2017.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Wolfram Challenges, Separate Ones by Zeroes.
Index entries for characteristic functions.

Cf. A000217, A002448, A005369, A023531, A035214, A022567, A052343, A006950, A106459, A127648.
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A106507 (reciprocal series).

(def A010054 (mapcat #(cons 1 (replicate % 0)) (range))) ; Tony Zorman, Apr 03 2023

a010054 = a010052 . (+ 1) . (* 8)
a010054_list = concatMap (\x -> 1 : replicate x 0) [0..]
-- Reinhard Zumkeller, Feb 12 2012, Oct 22 2011, Apr 02 2011

Basis( ModularForms( Gamma0(16), 1/2), 362) [2] ; /* Michael Somos, Jun 10 2014 */

A010054 := proc(n)
    if issqr(1+8*n) then
        1;
    else
        0;
    end if;
end proc:
seq(A010054(n),n=0..80) ; # R. J. Mathar, Feb 22 2021

a[ n_] := SquaresR[ 1, 8 n + 1] / 2; (* Michael Somos, Nov 15 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^2], {x, 0, n + Floor @ Sqrt[n]}] // Normal // TrigToExp) /. {y -> x}, {x, 0, n}]]; (* Michael Somos, Nov 15 2011 *)
Table[If[IntegerQ[(Sqrt[8n+1]-1)/2],1,0],{n,0,110}] (* Harvey P. Dale, Oct 29 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]; (* Michael Somos, Jul 01 2014 *)
Module[{tr=Accumulate[Range[20]]},If[MemberQ[tr,#],1,0]&/@Range[Max[tr]]] (* Harvey P. Dale, Mar 13 2023 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A), n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = issquare( 8*n + 1)}; /* Michael Somos, Apr 27 2000 */

a(n) = ispolygonal(n, 3); \\ Michel Marcus, Jan 22 2015

from sympy import integer_nthroot
def A010054(n): return int(integer_nthroot((n<<3)+1,2)[1]) # Chai Wah Wu, Nov 15 2022

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(-1, 0)
a = EulerTransform(b)
print([a(n) for n in range(88)]) # Peter Luschny, Nov 17 2022

Expansion of f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of q^(-1) * (phi(q) - phi(q^4)) / 2 in powers of q^8. - Michael Somos, Jul 01 2014
Expansion of q^(-1/8) * eta(q^2)^2 / eta(q) in powers of q. - Michael Somos, Apr 13 2005
Euler transform of period 2 sequence [ 1, -1, ...]. - Michael Somos, Mar 24 2003
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u6^3 + u2*u3^3 - u1*u2^2*u6. - Michael Somos, Apr 13 2005
a(n) = b(8*n + 1) where b()=A098108() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p > 2. - Michael Somos, Jun 06 2005
a(n) = A005369(2*n). - Michael Somos, Apr 29 2003
G.f.: theta_2(sqrt(q)) / (2 * q^(1/8)).
G.f.: 1 / (1 - x / (1 + x / (1 + x^1 / (1 - x / (1 + x / (1 + x^2 / (1 - x / (1 + x / (1 + x^3 / ...))))))))). - Michael Somos, May 11 2012
G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, May 02 2002
a(0)=1; for n>0, a(n) = A002024(n+1)-A002024(n). - Benoit Cloitre, Jan 05 2004
G.f.: Sum_{j>=0} Product_{k=0..j} x^j. - Jon Perry, Mar 30 2004
a(n) = floor((1-cos(Pi*sqrt(8*n+1)))/2). - Carl R. White, Mar 18 2006
a(n) = round(sqrt(2n+1)) - round(sqrt(2n)). - Hieronymus Fischer, Aug 06 2007
a(n) = ceiling(2*sqrt(2n+1)) - floor(2*sqrt(2n)) - 1. - Hieronymus Fischer, Aug 06 2007
a(n) = f(n,0) with f(x,y) = if x > 0 then f(x-y,y+1), otherwise 0^(-x). - Reinhard Zumkeller, Sep 27 2008
a(n) = A035214(n) - 1.
From Mikael Aaltonen, Jan 22 2015: (Start)
Since the characteristic function of s-gonal numbers is given by floor(sqrt(2n/(s-2) + ((s-4)/(2s-4))^2) + (s-4)/(2s-4)) - floor(sqrt(2(n-1)/(s-2) + ((s-4)/(2s-4))^2) + (s-4)/(2s-4)), by setting s = 3 we get the following: For n > 0, a(n) = floor(sqrt(2*n+1/4)-1/2) - floor(sqrt(2*(n-1)+1/4)-1/2).
(End)
a(n) = (-1)^n * A106459(n). - Michael Somos, May 04 2016
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(-1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A002448. - Michael Somos, May 05 2016
G.f.: Sum_{n >= 0} x^(n*(n+1)/2) = Product_{n >= 1} (1 - x^n)*(1 + x^n)^2 = Product_{n >= 1} (1 - x^(2*n))*(1 + x^n) = Product_{n >= 1} (1 - x^(2*n))/(1 - x^(2*n-1)). From the sum and product representations of theta_2(0, sqrt(q))/(2*q^(1/8)) function. The last product, given by Vladeta Jovovic above, is obtained from the second to last one by an Euler identity, proved via f(x) := Product_{n >= 1} (1 - x^(2*n-1))*Product_{n >= 1} (1 + x^n) = f(x^2), by moving the odd-indexed factors of the second product to the first product. This leads to f(x) = f(0) = 1. - Wolfdieter Lang, Jul 05 2016
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 08 2017
G.f.: Sum_{n >= 0} x^n * Product_{k >= n+1} (1 - x^(2*k)) = 1/(1 - x) * Sum_{n >= 0} x^(3*n) * Product_{k >= n+1} (1 - x^(2*k)) = 1/((1 - x)*(1 - x^3)) * Sum_{n >= 0} x^(5*n) * Product_{k >= n+1} (1 - x^(2*k)) = .... - Peter Bala, Jun 24 2025

Additional comments from Michael Somos, Apr 27 2000

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

Original entry on oeis.org

1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612, -370944, -577738, 401856, 1217160, 987136, -6905934, 2727432, 10661420, -7109760, -4219488, -12830688, 18643272, 21288960, -25499225, 13865712, -73279080, 24647168

Offset: 1

N. J. A. Sloane

Coefficients of the cusp form of weight 12 for the full modular group.
It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).
M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 (mod p) are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane, Mar 25 2007
Number 1 of the 74 eta-quotients listed in Table I of Martin (1996).
With Dedekind's eta function and the discriminant Delta one has eta(z)^24 = Delta(z)/(2*Pi)^12 = Sum_{m >= 1} tau(m)*q^m, with q = exp(2*Pi*i*z), and z in the complex upper half plane, where i is the imaginary unit. Delta is the eigenfunction of the Hecke operator T_n (n >= 1) with eigenvalue tau(n): T_n Delta = tau(n) Delta. From this the formula for tau(m)*tau(n) given below in the formula section follows. See, e.g., the Koecher-Krieg reference, Lemma and Satz, p. 212. Or the Apostol reference, eq. (3) on p. 114 and the first part of section 6.13 on p. 131. - Wolfdieter Lang, Jan 26 2016
For the functional equation satisfied by the Dirichlet series F(s), Re(s) > 7, of a(n) see the Hardy reference, p. 173, (10.9.4). It is (2*Pi)^(-s) * Gamma(s) * F(s) = (2*Pi)^(s-12) * Gamma(12-s) * F(12-s). This is attributed to J. R. Wilton, 1929, on p. 185. - Wolfdieter Lang, Feb 08 2017
Conjecture: |a(n)| with n > 1 can never be a perfect power. This has been verified for n up to 10^6. - Zhi-Wei Sun, Dec 18 2024
Conjecture: The numbers |a(n)| (n = 1,2,3,...) are distinct. This has been verified for the first 10^6 terms. - Zhi-Wei Sun, Dec 21 2024
Conjecture: |a(n)| > 2*n^4 for all n > 2. This has been verified for n = 3..10^6. - Zhi-Wei Sun, Dec 25 2024
Conjecture: a(m)^2 + a(n)^2 can never be a perfect power. This implies Lehmer's conjecture that a(n) is never zero. We have verified that there is no perfect power among a(m)^2 + a(n)^2 with m,n <= 1000 . - Zhi-Wei Sun, Dec 28 2024
Conjecture: The equation |a(m)a(n)| = x^k with m < n, k > 1 and x >= 0 has no solution. This has been verified for m < n <= 5000. - Zhi-Wei Sun, Dec 29 2024
For some conjectures motivated by additive combinatorics, one may consult the link to Question 485138 at MathOverflow. - Zhi-Wei Sun, Jan 25 2025

			G.f. = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + ...
35328 = (-24)*(-1472) = a(2)*a(4) = a(2*4) + 2^11*a(2*4/4) = 84480 + 2048*(-24) = 35328. See a comment on T_n Delta = tau(n) Delta above. - _Wolfdieter Lang_, Jan 21 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, lecture X, pp. 161-185.
Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.
Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, p. 139.
M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, see p. 482.
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).
H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.

Simon Plouffe, Table of n, a(n) for n = 1..16090
Jennifer S. Balakrishnan, William Craig, and Ken Ono, Variations of Lehmer's Conjecture for Ramanujan's tau-function, arXiv:2005.10345 [math.NT], 2020.
Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai, Even values of Ramanujan's tau-function, arXiv:2102.00111 [math.NT], 2021.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024.
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory, Vol. 98, No. 2 (2003), 377-389. MR1955423 (2003k:11071).
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Benoit Cloitre, On the fractal behavior of primes, 2011.
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Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(l), arXiv:1312.6819 [math.NT], 2013-2014.
Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman, Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2010.
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Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan's tau function, Discrete Mathematics, Vol. 341, No. 10 (2018), pp. 2831-2840; arXiv preprint, arXiv:1606.08037 [math.CO], 2016.
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James Lee Hafner and Jeffrey Stopple, A Heat Kernel Associated to Ramanujan's Tau Function, The Ramanujan Journal, Vol. 4, No. 2 (2000), pp. 123-128.
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D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 (1970), 495-496.
LMFDB, Newform orbit 1.12.a.a
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Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
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Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
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M. Ram Murty, V. Kumar Murty, and T. N. Shorey, Odd values of the Ramanujan tau-function, Bulletin de la S. M. F., tome 115 (1987), p. 391-395.
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Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's tau function, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 9, no 1 (1967-1968), exp. no 14, pp. 1-17.
Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's Tau function, 1997.
Jean-Pierre Serre, Sur la lacunarité des puissances de eta, Glasgow Math. Journal, 27 (1985), 203-221.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, My Favorite Integer Sequences, arXiv:math/0207175 [math.CO], 2002.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
David A. Steffen, Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n), 1998.
William A. Stein, Database.
Zhi-Wei Sun, Questions on the reciprocals of the values of the tau function, Question 484763 at MathOverflow, December 25, 2024.
Zhi-Wei Sun, Pythagorean triples and Ramanujan's tau function, Question 484936 at MathOverflow, December 28, 2024.
Zhi-Wei Sun, Additive combinatorics for Ramanujan's tau function, Question 485138 at MathOverflow, January 1, 2025.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
F. Van der Blij, The function tau (n) of S. Ramanujan (an expository lecture), Math. Student, Vol. 18, No. 3 (1950), pp. 83-99; entire issues 3 and 4.
Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
G. N. Watson, A table of Ramanujan's function tau(n), Proc. London Math. Soc., 51 (1950), 1-13.
Eric Weisstein's World of Mathematics, Tau Function.
Kenneth S. Williams, Historical remark on Ramanujan's tau function, Amer. Math. Monthly, 122 (2015), 30-35; author's copy.
Index entries for "core" sequences.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m.
Index entries for sequences related to Chebyshev polynomials.

Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).
For a(n) mod N for various values of N see A046694, A098108, A126812-...
Cf. A006352, A099059, A099060, A262339, A292781.
For primes p such that tau(p) == -1 (mod 23) see A106867.
Cf. A010816, A004009. A013973.
Cf. A126832(n) = a(n) mod 5.

using Nemo
function DedekindEta(len, r)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(r, len, z)
    [coeff(e, j) for j in 0:len - 1] end
RamanujanTauList(len) = DedekindEta(len, 24)
RamanujanTauList(28) |> println # Peter Luschny, Mar 09 2018

M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);

Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */

M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);

CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* Dean Hickerson, Jan 03 2003 *)
max = 28; g[k_] := -BernoulliB[k]/(2k) + Sum[ DivisorSigma[k - 1, n - 1]*q^(n - 1), {n, 2, max + 1}]; CoefficientList[ Series[ 8000*g[4]^3 - 147*g[6]^2, {q, 0, max}], q] // Rest (* Jean-François Alcover, Oct 10 2012, from modular forms *)
RamanujanTau[Range[40]] (* The function RamanujanTau is now part of Mathematica's core language so there is no longer any need to load NumberTheory`Ramanujan` before using it *) (* Harvey P. Dale, Oct 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
a[ n_] := With[{t = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ DedekindEta[t]^24, {q, 0, n}], {q, 0, n}]]; (* Michael Somos, May 27 2014 *)

{a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};

{a(n) = if( n<1, 0, polcoeff( x * (sum( i=1, (sqrtint( 8*n - 7) + 1) \ 2,(-1)^i * (2*i - 1) * x^((i^2 - i)/2), O(x^n)))^8, n))};

taup(p,e)={
    if(e==1,
        (65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
    ,
        my(t=taup(p,1));
        sum(j=0,e\2,
            (-1)^j*binomial(e-j,e-2*j)*p^(11*j)*t^(e-2*j)
        )
    )
};
a(n)=my(f=factor(n));prod(i=1,#f[,1],taup(f[i,1],f[i,2]));
\\ Charles R Greathouse IV, Apr 22 2013

\\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):
a(n) = n^4*sigma(n) - 24*sum(k=1, n-1, (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k));
vector(33, n, a(n)) \\ Joerg Arndt, Sep 06 2015

a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A000594(n): return n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) # Chai Wah Wu, Nov 08 2022

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A000594(n)
  ary = [1]
  a = [0] + (1..n - 1).map{|i| s(i)}
  (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
  ary
end
p A000594(100) # Seiichi Manyama, Mar 26 2017

def A000594(n)
  ary = [0, 1]
  (2..n).each{|i|
    s, t, u = 0, 1, 0
    (1..n).each{|j|
      t += 9 * j
      u += j
      break if i <= u
      s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u]
    }
    ary << s / (i - 1)
  }
  ary[1..-1]
end
p A000594(100) # Seiichi Manyama, Nov 25 2017

CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013

list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016

G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 04 2011
abs(a(n)) = O(n^(11/2 + epsilon)), abs(a(p)) <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.
Zagier says: The proof of these formulas, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u + 48*v + 4096*w) - v^3. - Michael Somos, Jul 19 2004
G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - Michael Somos, Dec 09 2013
a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - Michael Somos, Apr 17 2015
a(n) = tau(n) (with tau(0) = 0): tau(m)*tau(n) = Sum_{d| gcd(m,n)} d^11*tau(m*n/d^2), for positive integers m and n. If gcd(m,n) = 1 this gives the multiplicativity of tau. See a comment above with the Koecher-Krieg reference, p. 212, eq. (5). - Wolfdieter Lang, Jan 21 2016
Dirichlet series as product: Sum_{n >= 1} a(n)/n^s = Product_{n >= 1} 1/(1 - a(prime(n))/prime(n)^s  + prime(n)^(11-2*s)). See the Mordell link, eq. (2). - Wolfdieter Lang, May 06 2016. See also Hardy, p. 164, eqs. (10.3.1) and (10.3.8). - Wolfdieter Lang, Jan 27 2017
a(n) is multiplicative with a(prime(n)^k) = sqrt(prime(n)^(11))^k*S(k, a(n) / sqrt(prime(n)^(11))), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2, and A076847(n) = a(prime(n)). See A076847 for alpha multiplicativity and examples. - Wolfdieter Lang, May 17 2016. See also Hardy, p. 164, eq. (10.3.6) rewritten in terms of S. - Wolfdieter Lang, Jan 27 2017
G.f. eta(z)^24 (with q = exp(2*Pi*i*z)) also (E_4(q)^3 - E_6(q)^2) / 1728. See the Hardy reference, p. 166, eq. (10.5.3), with Q = E_4 and R = E_6, given in A004009 and A013973, respectively. - Wolfdieter Lang, Jan 30 2017
a(n) (mod 5) == A126832(n).
a(1) = 1, a(n) = -(24/(n-1))*Sum_{k=1..n-1} A000203(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 26 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Euler Transform of [-24, -24, -24, -24, ...]. - Simon Plouffe, Jun 21 2018
a(n) = n^4*sigma(n)-24*Sum_{k=1..n-1} (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k). [See Douglas Niebur link]. - Wesley Ivan Hurt, Jul 22 2025

A005369 a(n) = 1 if n is of the form m(m+1), else 0.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

This is essentially the q-expansion of the Jacobi theta function theta_2(q). (In theta_2 one has to ignore the initial factor of 2*q^(1/4). See also A010054.) - N. J. A. Sloane, Aug 03 2014

For n > 0, a(n) is the number of partitions of n into two parts such that the larger part is equal to the square of the smaller part. - Wesley Ivan Hurt, Dec 23 2020

			G.f. = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 + x^56 + x^72 + x^90 + ...
G.f. = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + ...

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Robert Price, Comments on A005369 concerning Elementary Cellular Automata, Jan 29 2016
Eric Weisstein's World of Mathematics, Jacobi Theta Functions [From _Franklin T. Adams-Watters_, Jun 29 2009]
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for characteristic functions

Cf. A002378. Partial sums give A000194.

Cf. A010052, A010054, A016813, A240025.

a005369 = a010052 . (+ 1) . (* 4) -- Reinhard Zumkeller, Jul 05 2014

A005369 := proc(n)
    if issqr(1+4*n) then
        if type( sqrt(1+4*n)-1,'even') then
            1;
        else
            0;
        end if;
    else
        0;
    end if;
end proc:
seq(A005369(n),n=0..80) ; # R. J. Mathar, Feb 22 2021

a005369[n_] := If[IntegerQ[Sqrt[4 # + 1]], 1, 0] & /@ Range[0, n]; a005369[100] (* Michael De Vlieger, Jan 02 2015 *)
a[ n_] := SquaresR[ 1, 4 n + 1] / 2; (* Michael Somos, Feb 22 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
QP = QPochhammer; s = QP[q^4]^2/QP[q^2] + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)
nmax = 200; CoefficientList[Series[Sum[x^(k*(k + 1)), {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2020 *)

{a(n) = if( n<0, 0, issquare(4*n + 1))};

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

from sympy.ntheory.primetest import is_square
def A005369(n): return int(is_square((n<<2)|1)) # Chai Wah Wu, Jun 07 2025

Expansion of q^(-1/4) * eta(q^4)^2 / eta(q^2) in powers of q.
Euler transform of period 4 sequence [ 0, 1, 0, -1, ...].
G.f.: Product_{k>0} (1 - x^(4*k)) / (1 - x^(4*k-2)) = f(x^2, x^6) where f(, ) is Ramanujan's general theta function.
From Michael Somos, Apr 13 2005: (Start)
Given g.f. A(x), then B(q) = (q*A(q^4))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 4*v*w^2 - u^2*w.
Given g.f. A(x), then B(q) = q*A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2^2*u6 - u1*u6^3 - u3^3*u2. (End)
a(n) = b(4*n + 1) where b() = A098108() is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p>2. - Michael Somos, Jun 06 2005
G.f.: 1/2 x^{-1/4}theta_2(0,x), where theta_2 is a Jacobi theta function. - Franklin T. Adams-Watters, Jun 29 2009
a(A002378(n)) = 1; a(A078358(n)) = 0. - Reinhard Zumkeller, Jul 05 2014
a(n) = floor(sqrt(n+1)+1/2)-floor(sqrt(n)+1/2). - Mikael Aaltonen, Jan 02 2015
a(2*n) = A010054(n).
a(n) = A000729(n)(mod 2). - John M. Campbell, Jul 16 2016
For n > 0, a(n) = Sum_{k=1..floor(n/2)} [k^2 = n-k], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Dec 23 2020

Additional comments from Michael Somos, Apr 29 2003

Erroneous formula removed by Reinhard Zumkeller, Jul 05 2014

A046694 Ramanujan tau numbers mod 691 = sum of 11th power of divisors mod 691.

Original entry on oeis.org

1, 667, 252, 601, 684, 171, 531, 178, 372, 168, 469, 123, 629, 385, 309, 388, 611, 55, 672, 630, 449, 491, 92, 632, 57, 106, 88, 580, 173, 185, 366, 666, 27, 538, 429, 379, 622, 456, 269, 136, 87, 280, 36, 632, 160, 556, 435, 345, 194, 14, 570, 52, 209, 652, 172, 542, 49

Offset: 1

Marc LeBrun

Ramanujan tau is multiplicative, so this sequence is multiplicative mod 691.
There are pairs of identical terms a(n) and a(n+1). The first such twin pair is a(184) = a(185) = 483. The indices for a first twin in a pair are listed in A121733. Corresponding twin values are listed in A121734. - Alexander Adamchuk, Aug 18 2006
Set of values of a(n) consists of all integers from 0 to 690. The first a(n) = 0 occur at n = 2*691 - 1 = 1381 that is a prime. Set of numbers n such that a(n) = 0 is a union of all terms of the arithmetic progressions k*p, where p is a prime of the form p = 2m*691 - 1 and k>0 is an integer. Primes of the form p = 2m*691 - 1 are listed in A134671 = {1381,5527,8291,12437,22111,29021,30403,...}. It appears that in a(n) there are strings of consecutive zeros of any length. The first pair of consecutive zeros occurs at n = {16581,16582}. The least numbers k such that a(n) has a string of n consecutive zeros starting with a(k) are listed in A134670(n) = {1381,16581,290217,1409635,...}. - Alexander Adamchuk, Nov 05 2007

			Coefficient of x^2 in tau(x) = -24; 1^11+2^11 = 2049 = 667 mod 691 = -24 mod 691.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 169, (10.6.4).

T. D. Noe, Table of n, a(n) for n = 1..10000
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A013959, A121733, A121734, A098108, A126812-...

Cf. also A134670, A134671, A121742, A121743, A262339.

A046694 := proc(n)
    numtheory[sigma][11](n) mod 691 ;
end proc: # R. J. Mathar, Feb 01 2013

a[n_] := Mod[Total[Divisors[n]^11], 691]; a /@ Range[57] (* Jean-François Alcover , Apr 22 2011 *)
Table[Mod[DivisorSigma[11,n],691],{n,60}] (* Harvey P. Dale, Jun 01 2012 *)

a(n)=ramanujantau(n)%691 \\ Charles R Greathouse IV, Feb 08 2017

a(n)=sigma(n,11)%691 \\ Charles R Greathouse IV, Sep 09 2022

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 11)%691 # Indranil Ghosh, Apr 24 2017

a(n) = A000594(n) mod A262339(6). - Jonathan Sondow, Sep 22 2015

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 5, 3, 8, 3, 5, 9, 10, 3, 8, 3, 8, 9, 5, 3, 11, 7, 5, 12, 8, 3, 13, 3, 14, 9, 5, 9, 15, 3, 5, 9, 11, 3, 13, 3, 8, 16, 5, 3, 17, 7, 8, 9, 8, 3, 11, 9, 11, 9, 5, 3, 18, 3, 5, 16, 19, 9, 13, 3, 8, 9, 13, 3, 20, 3, 5, 16, 8, 9, 13, 3, 17, 21, 5, 3, 18, 9, 5, 9, 11, 3, 18, 9, 8, 9, 5, 9, 22, 3, 8, 16, 15, 3, 13, 3, 11, 23, 5, 3, 20, 3, 13

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1, and for n > 1, b(n) = A046523(n) + A000035(n), which starts as 1, 2, 3, 4, 3, 6, 3, 8, 5, 6, 3, 12, 3, 6, 7, 16, 3, 12, 3, 12, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A291767, A291768 (bisections), A147516.
Cf. A046523, A101296, A286161, A286251, A286367, A291762 (related or similar filtering sequences).
Cf. A065091 (positions of 3's), A100484 (of 4 and 5's), A001248 (of 4 and 7's), A046388 (of 9's), A030078 (of 6 and 12's).
Cf. A098108 (one of the matching sequences).

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
write_to_bfile(1,rgs_transform(vector(65537,n,((-1)^n)*A046523(n))),"b291761.txt");
\\ Or alternatively:
f(n) = if(1==n,n,A046523(n)+(n%2));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291761.txt");

A018255 Divisors of 30.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 15, 30

Offset: 1

N. J. A. Sloane

For n > 1, These are also numbers m such that k^4 + (k+1)^4 + ... + (k + m - 1)^4 is prime for some k and numbers m such that k^8 + (k+1)^8 + ... + (k + m - 1)^8 is prime for some k. - Derek Orr, Jun 12 2014
These seem to be the numbers m such that tau(n) = n*sigma(n) mod m for all n. See A098108 (mod 2), A126825 (mod 3), and A126832 (mod 5). - Charles R Greathouse IV, Mar 17 2022
The squarefree 5-smooth numbers: intersection of A051037 and A005117. - Amiram Eldar, Sep 26 2023

			From the second comment: 1^3 + 2^3 + 2^3 + 2^3 + 4^3 + 4^3 + 4^3 + 8^3 = (1 + 2 + 2 + 2 + 4 + 4 + 4 + 8)^2 = 729. - _Bruno Berselli_, Dec 28 2014

Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, C. Scribner's Sons (1972), Chapter XIII, Paragraph 349.

Edward Barbeau and Samer Seraj, Sum of Cubes is Square of Sum, arXiv:1306.5257 [math.NT], 2013.
Index entries for sequences related to divisors of numbers.

Cf. A000005, A005117, A051037, A158649, A161715.

Cf. A098108, A126825, A126832.

Divisors(30); // Bruno Berselli, Dec 28 2014

Divisors[30] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)

PARI
```
divisors(30)
```

a(n) = A161715(n-1). - Reinhard Zumkeller, Jun 21 2009

Sum_{i=1..8} A000005(a(i))^3 = (Sum_{i=1..8} A000005(a(i)))^2, see Kordemsky in References and Barbeau et al. in Links section. - Bruno Berselli, Dec 28 2014

A033683 a(n) = 1 if n is an odd square not divisible by 3, otherwise 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

			G.f. = x + x^25 + x^49 + x^121 + x^169 + x^289 + x^361 + x^529 + x^625 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 105, Eq. (41).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions.

Cf. A098108, A033684.

Cf. A010052, A010872, A104777.

a033683 n = fromEnum $ odd n && mod n 3 > 0 && a010052 n == 1
-- Reinhard Zumkeller, Nov 14 2015

Basis( ModularForms( Gamma0(144), 1/2), 106)[2]; /* Michael Somos, Dec 07 2019 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^4] - EllipticTheta[ 2, 0, x^36])/2, {x, 0, n}] // PowerExpand; (* Michael Somos, Dec 07 2019 *)
Table[If[OddQ[n]&&IntegerQ[Sqrt[n]]&&Mod[n,3]!=0,1,0],{n,0,120}] (* Harvey P. Dale, Sep 06 2020 *)

{a(n) = if( n%24 == 1, issquare(n), 0)}; /* Michael Somos, Jan 26 2008 */

Essentially the series psi_6(z)=(1/2)(theta_2(z/9)-theta_2(z)).
a(A104777(n)) = 1.
A080995(n) = a(24n+1).
Multiplicative with a(p^e) = 1 if 2 divides e and p > 3, 0 otherwise. - Mitch Harris, Jun 09 2005
Euler transform of a period 144 sequence. - Michael Somos, Jan 26 2008
a(n) = A033684(n) * A000035(n).
Dirichlet g.f.: zeta(2*s) *(1-2^(-2s)) *(1-3^(-2s)). - R. J. Mathar, Mar 10 2011
G.f.: Sum_{k in Z} x^(6*k+1)^2. - Michael Somos, Dec 07 2019
Sum_{k=1..n} a(k) ~ sqrt(n)/3. - Amiram Eldar, Jan 14 2024

A126812 Ramanujan numbers (A000594) read mod 4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

D. B. Lahiri, On Ramanujan's function tau(n) and divisor function sigma_k(n), I, Bulletin of the Calcutta Mathematical Society, Vol. 38 (1946), pp. 193-206; II, ibid., Vol. 39 (1947), pp. 33-51.

Antti Karttunen, Table of n, a(n) for n = 1..65537
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A008442, A013955, A098108, A126813, A126814.

Mod[#, 4] & /@ RamanujanTau@ Range@ 105 (* Michael De Vlieger, Nov 26 2017 *)

A126812(n) = (ramanujantau(n)%4); \\ Antti Karttunen, Nov 26 2017

a(n) == n^2 * sigma_7(n) (mod 4) (Lahiri, 1946-1947). - Amiram Eldar, Jan 04 2025

A326306 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - p^(1 - s) + p^(-s)).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 5, 4, 2, 8, 2, 4, 4, 16, 2, 10, 2, 8, 4, 4, 2, 16, 7, 4, 14, 8, 2, 8, 2, 32, 4, 4, 4, 20, 2, 4, 4, 16, 2, 8, 2, 8, 10, 4, 2, 32, 9, 14, 4, 8, 2, 28, 4, 16, 4, 4, 2, 16, 2, 4, 10, 64, 4, 8, 2, 8, 4, 8, 2, 40, 2, 4, 14, 8, 4, 8, 2, 32, 41, 4, 2, 16, 4

Offset: 1

Ilya Gutkovskiy, Oct 17 2019

Inverse Moebius transform of A003557.

Dirichlet convolution of A000203 with A097945.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A000010, A000079 (fixed points), A000203, A003557, A007947, A008683, A098108 (parity of a(n)), A191750, A300717, A335032.

Table[Sum[d/Last[Select[Divisors[d], SquareFreeQ]], {d, Divisors[n]}], {n, 1, 85}]
Table[Sum[MoebiusMu[n/d] EulerPhi[n/d] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 85}]
f[p_, e_] := 1 + (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

G.f.: Sum_{k>=1} (k / rad(k)) * x^k / (1 - x^k), where rad = A007947.
a(n) = Sum_{d|n} A003557(d).
a(n) = Sum_{d|n} mu(n/d) * phi(n/d) * sigma(d), where mu = A008683, phi = A000010 and sigma = A000203.
a(p) = 2, where p is prime.
From Vaclav Kotesovec, Jun 20 2020: (Start)
Dirichlet g.f.: zeta(s) * Product_{primes p} (1 + 1/(p^s - p)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) + p^(-s)). (End)
Multiplicative with a(p^e) = 1 + (p^e-1)/(p-1). - Amiram Eldar, Oct 14 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*sigma(gcd(n,k)).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

cp's OEIS Frontend

A126811 Duplicate of A098108.

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A010054 a(n) = 1 if n is a triangular number, otherwise 0.

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A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

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A005369 a(n) = 1 if n is of the form m(m+1), else 0.

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A046694 Ramanujan tau numbers mod 691 = sum of 11th power of divisors mod 691.

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