A013959 -id:A013959 - cp's OEIS frontend

A000156 Number of ways of writing n as a sum of 24 squares.

1, 48, 1104, 16192, 170064, 1362336, 8662720, 44981376, 195082320, 721175536, 2319457632, 6631997376, 17231109824, 41469483552, 93703589760, 200343312768, 407488018512, 793229226336, 1487286966928, 2697825744960, 4744779429216

Offset: 0

N. J. A. Sloane

The Carlitz paper has the wrong formula on p. 505, eq. (3). The factor in front of tau(n/2) should be -2^16 (not -2^12). The mistake appeared in the previous equation derived from eq. (2) where theta_3^(24) * 256*k^4*k'^4 was replaced by 2^8*g(q^2) which produces the factor 2^8*256 = 2^16. (One should subtract on p. 504 the second equation in the middle from the negative of the first equation. There is also a sign mistake in the sum term of the third equation from the bottom.) - Wolfdieter Lang, Sep 24 2016

Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 195, eq. (15.1).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 107.
G. H. Hardy, Ramanujan, 1940, Cambridge, reprinted with additional corrections and comments by AMS Chelsea Publishing, 1999, 2002, Providence, Rhode Island, ch. IX., pp. 153-155.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Carlitz, On the representation of an integer as the sum of twenty-four squares, Indagationes Mathematicae (Proceedings), 58 (1955) 504-506.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Index entries for sequences related to sums of squares

Row d=24 of A122141 and of A319574, 24th column of A286815.

(sum(x^(m^2),m=-10..10))^24; seq(coeff(%,x,n), n=0..30);
# Alternative:
A000156list := proc(len) series(JacobiTheta3(0, x)^24, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000156list(21); # Peter Luschny, Oct 02 2018

Table[SquaresR[24, n], {n, 0, 20}] (* Ray Chandler, Nov 28 2006 *)

first(n)=my(x='x); x+=O(x^(n+1)); Vec((2*sum(k=1,sqrtint(n),x^k^2) + 1)^24) \\ Charles R Greathouse IV, Jul 29 2016

From Wolfdieter Lang, Sep 24 2016: (Start)
For n >= 1: a(n) = (16*sigma^*{11} - 128*(512*tau(n/2) + (-1)^n*259*tau(n)))/691, with sigma^*{11} = sigma_{11}^{e}(n) - sigma_{11}^{o}(n) if n even and sigma_{11}(n) otherwise. Here sigma_{11}(n) = A013959(n) and 0 if n is not an integer, sigma_{11}^{e}(n) and sigma_{11}^{o}(n) are the sums of the 11th power of the odd and even positive divisors of n, respectively. Ramanujan's tau(n) = A000594(n) and 0 if n is not an integer. This is from Hardy, ch. IX., p. 155, eqs. (9.17.1) and (9.17.2), and p.142 for the definition of sigma^*_{nu}(n). See also the Ash and Gross reference.
Another version, see the corrected Carlitz reference:
a(n) = (2^4*(sigma_{11}(n)- 2*sigma_{11}(n/2) + 2^{12}*sigma_{11}(n/4)) - 2^7*259*(-1)^n*tau(n) - 2^16*tau(n/2))/691, n >= 1.
(End)
a(n) = (48/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 4098, 531444, 16785412, 244140630, 2177857512, 13841287208, 68753047560, 282431130813, 1000488301740, 3138428376732, 8920506494928, 23298085122494, 56721594978384, 129747072969720, 281612482805776, 582622237229778, 1157402774071674

Offset: 0

Seiichi Manyama, Feb 18 2017

Multiplicative because A013959 is. - Andrew Howroyd, Jul 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), this sequence (phi_{12, 1}).
Cf. A282549 (E_2*E_4^3), A282576 (E_2*E_6^2), A058550 (E_14).
Cf. A013670.

Table[n * DivisorSigma[11, n], {n, 0, 18}] (* Amiram Eldar, Sep 06 2023 *)

a(n) = if(n < 1, 0, n*sigma(n, 11)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n*A013959(n) for n > 0.
a(n) = (441*A282549(n) + 250*A282576(n) - 691*A058550(n))/65520.
Sum_{k=1..n} a(k) ~ zeta(12) * n^13 / 13. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(11*e+11)-1)/(p^11-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-12). (End)

A321815 Sum of 11th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 177148, 1, 48828126, 177148, 1977326744, 1, 31381236757, 48828126, 285311670612, 177148, 1792160394038, 1977326744, 8649804864648, 1, 34271896307634, 31381236757, 116490258898220, 48828126, 350279478046112, 285311670612, 952809757913928

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Eric Weisstein's World of Mathematics, Odd Divisor Function.
Index entries for sequences mentioned by Glaisher.

Column k=11 of A285425.
Cf. A050999, A051000, A051001, A051002, A321810 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013670, A013959.

List(List(List([1..25],j->DivisorsInt(j)),i->Filtered(i,k->IsOddInt(k))),m->Sum(m,n->n^11)); # Muniru A Asiru, Dec 07 2018

a[n_] := DivisorSum[n, #^11&, OddQ[#]&]; Array[a, 20] (* Amiram Eldar, Dec 07 2018 *)

apply( A321815(n)=sigma(n>>valuation(n,2),11), [1..30]) \\ M. F. Hasler, Nov 26 2018

from sympy import divisor_sigma
def A321815(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),11)) # Chai Wah Wu, Jul 16 2022

a(n) = A013959(A000265(n)) = sigma_11(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^11*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(11*e+11)-1)/(p^11-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^12, where c = zeta(12)/24 = 691*Pi^12/15324309000 = 0.0416769... . (End)

A055715 Numbers k such that k | sigma_11(k).

Original entry on oeis.org

1, 6, 28, 120, 402, 496, 644, 672, 920, 1366, 1608, 1932, 2680, 2760, 3417, 3966, 4098, 4623, 4975, 5152, 6210, 6834, 8040, 8128, 8280, 9246, 9528, 9950, 12294, 13668, 15008, 15456, 15864, 16392, 18492, 19900, 24120, 24840, 25954, 27320, 27336, 29850, 30240, 32760

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_11(k) is the sum of the 11th powers of the divisors of k (A013959).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A013959.

Cf. A007691, A046762, A046763, A046764, A055709, A055710, A055711, A055712, A055713, A055714.

Do[If[Mod[DivisorSigma[11, n], n]==0, Print[n]], {n, 1, 40000}]

isok(k) = (sigma(k, 11) % k) == 0; \\ Michel Marcus, Nov 09 2019

a(37)-a(40) corrected and more terms added by Amiram Eldar, Nov 09 2019

A126846 Ramanujan numbers (A000594) read mod 23^2.

Original entry on oeis.org

1, 505, 252, 115, 69, 300, 184, 369, 92, 460, 322, 414, 459, 345, 460, 22, 161, 437, 483, 0, 345, 207, 254, 413, 162, 93, 116, 0, 344, 69, 229, 230, 207, 368, 0, 0, 207, 46, 346, 69, 160, 184, 138, 0, 0, 252, 459, 254, 139, 344, 368, 414, 253, 390, 0, 184, 46, 208, 71, 0

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Pierre Serre, Une interprétation des congruences relatives à la fonction tau de Ramanujan, Séminaire Delange-Pisot-Poitou, Théorie des nombres, Vol. 9, No. 1 (1967-1968), Talk no. 14, 17 p., section 4.5, page 14-11.
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.

a[n_] := Mod[RamanujanTau[n], 529]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 529; \\ Amiram Eldar, Jan 05 2025

a(p) == sigma_11(p) (mod 23^2) for prime p of the form u^2 + 23*v^2, u >= 1 (Serre, 1968). - Amiram Eldar, Jan 05 2025

A055705 Numbers k such that k | sigma_11(k) - phi(k)^11.

Original entry on oeis.org

1, 2, 10, 12, 82, 168, 952, 1716, 2732, 2970, 5627, 8185, 11400, 12871, 20104, 20368, 23526, 25749, 70176, 82920, 111194, 117151, 119160, 128790, 134670, 143136, 185140, 193020, 208352, 240408, 247995, 251856, 291368, 354588, 565768, 592006, 642600, 783315

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_11(k) is the sum of the 11th powers of the divisors of k (A013959).

Amiram Eldar, Table of n, a(n) for n = 1..124

Cf. A000010, A013959.

Do[If[Mod[DivisorSigma[11, n]-EulerPhi[n]^11, n]==0, Print[n]], {n, 10^5}]
Select[Range[800000],Divisible[DivisorSigma[11,#]-EulerPhi[#]^11,#]&] (* Harvey P. Dale, Apr 15 2018 *)

isok(n) = !((sigma(n, 11) - eulerphi(n)^11) % n); \\ Michel Marcus, Mar 02 2014

Definition corrected and more terms from Michel Marcus, Mar 02 2014

A126818 Ramanujan numbers (A000594) read mod 256.

Original entry on oeis.org

1, 232, 252, 64, 222, 96, 152, 0, 21, 48, 84, 0, 54, 192, 136, 0, 178, 8, 44, 128, 160, 32, 72, 0, 167, 240, 152, 0, 102, 64, 96, 0, 176, 80, 208, 64, 62, 224, 40, 0, 122, 0, 180, 0, 54, 64, 16, 0, 169, 88, 56, 128, 110, 192, 216, 0, 80, 112, 228, 0, 198, 0, 120, 0, 212, 128, 188

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and Bruce C. Berndt, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions, in: Ramanujan's Lost Notebook, Part III, Springer, New York, NY, 2012.
R. P. Bambah and S. Chowla, The Residue of Ramanujan's Function tau(n) to the Modulus 2^8, Journal of the London Mathematical Society, Vol s1-22, No. 2 (1947), pp. 140-147.
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.

a[n_] := Mod[RamanujanTau[n], 256]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 256; \\ Amiram Eldar, Jan 05 2025

a(n) == sigma_11(n) (mod 256) for n odd (Bambah and Chowla, 1947; Andrews and Berndt, 2012, eq. (5.12.26), p. 118). - Amiram Eldar, Jan 05 2025

A126821 Ramanujan numbers (A000594) read mod 2048.

Original entry on oeis.org

1, 2024, 252, 576, 734, 96, 1688, 512, 1045, 816, 84, 1792, 1846, 448, 648, 0, 1970, 1544, 1580, 896, 1440, 32, 328, 0, 423, 752, 408, 1536, 1126, 832, 1376, 0, 688, 1872, 2000, 1856, 1342, 992, 296, 1024, 890, 256, 1716, 1280, 1078, 320, 1808, 0, 1449, 88, 824, 384

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and Bruce C. Berndt, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions, in: Ramanujan's Lost Notebook, Part III, Springer, New York, NY, 2012.
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.

a[n_] := Mod[RamanujanTau[n], 2048]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 2048; \\ Amiram Eldar, Jan 05 2025

From Amiram Eldar, Jan 05 2025: (Start)
a(n) == sigma_11(n) (mod 2048) for n == 1 (mod 8) (Kolberg, 1962).
a(n) == 24 * sigma_11(n) (mod 2048) (Andrews and Berndt, 2012, p. 118). (End)

A126822 Ramanujan numbers (A000594) read mod 4096.

Original entry on oeis.org

1, 4072, 252, 2624, 734, 2144, 3736, 2560, 1045, 2864, 2132, 1792, 3894, 448, 648, 0, 4018, 3592, 3628, 896, 3488, 2080, 2376, 2048, 2471, 752, 2456, 1536, 1126, 832, 3424, 0, 688, 1872, 2000, 1856, 1342, 3040, 2344, 3072, 2938, 2304, 3764, 3328, 1078, 320

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Amiram Eldar, 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.

a[n_] := Mod[RamanujanTau[n], 4096]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 4096; \\ Amiram Eldar, Jan 05 2025

a(n) == 1537 * sigma_11(n) (mod 4096) for n == 5 (mod 8) (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

A126823 Ramanujan numbers (A000594) read mod 8192.

Original entry on oeis.org

1, 8168, 252, 6720, 4830, 2144, 7832, 2560, 1045, 6960, 2132, 5888, 3894, 448, 4744, 4096, 8114, 7688, 3628, 896, 7584, 6176, 6472, 6144, 2471, 4848, 6552, 5632, 5222, 832, 3424, 0, 4784, 1872, 6096, 1856, 1342, 3040, 6440, 3072, 2938, 6400, 3764, 7424, 1078

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Amiram Eldar, 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.

a[n_] := Mod[RamanujanTau[n], 8192]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 8192; \\ Amiram Eldar, Jan 05 2025

a(n) == 1217 * sigma_11(n) (mod 8192) for n == 3 (mod 8) (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

cp's OEIS Frontend

A000156 Number of ways of writing n as a sum of 24 squares.

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A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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A321815 Sum of 11th powers of odd divisors of n.

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A055715 Numbers k such that k | sigma_11(k).

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A126846 Ramanujan numbers (A000594) read mod 23^2.

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A055705 Numbers k such that k | sigma_11(k) - phi(k)^11.

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A126818 Ramanujan numbers (A000594) read mod 256.

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