A000594 -id:A000594 - cp's OEIS frontend

A099059 The odd bisection of A000594.

1, 252, 4830, -16744, -113643, 534612, -577738, 1217160, -6905934, 10661420, -4219488, 18643272, -25499225, -73279080, 128406630, -52843168, 134722224, -80873520, -182213314, -145589976, 308120442, -17125708, -548895690, 2687348496, -1696965207, -1740295368, -1596055698

Offset: 0

N. J. A. Sloane, Nov 15 2004

sign

			G.f. = 1 + 252*x + 4830*x^2 - 16744*x^3 - 113643*x^4 + 534612*x^5 + ...
G.f. = q + 252*q^3 + 4830*q^5 - 16744*q^7 - 113643*q^9 + 534612*q^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mathew Rogers, Identities for the Ramanujan zeta function, arXiv:1210.1942 [math.NT], 2012-2013. See page 3 Lemma 1.

Cf. A000594, A099060.

a[ n_] := SeriesCoefficient[ q QPochhammer[q]^24, {q, 0, 2 n + 1}]; (* Michael Somos, Apr 17 2015 *)
a[n_] := RamanujanTau[2*n+1]; Array[a, 30, 0] (* Amiram Eldar, Jan 10 2025 *)

{a(n) = if( n<0, 0, n = 2*n+1; polcoeff( x * eta(x + x * O(x^n))^24, n))}; /* Michael Somos, Apr 17 2015 */

Expansion of (F(q) - F(-q)) / 2 = F(q) + 24*F(q^2) + 2048*F(q^4) in powers of q^2 where F() is the g.f. of A000594. - Michael Somos, Apr 17 2015

a(n) = A000594(2*n + 1). - Michael Somos, Apr 17 2015

More terms from Joshua Zucker, May 15 2006

A099060 The even bisection of A000594.

Original entry on oeis.org

-24, -1472, -6048, 84480, -115920, -370944, 401856, 987136, 2727432, -7109760, -12830688, 21288960, 13865712, 24647168, -29211840, -196706304, 165742416, 167282496, -255874080, 408038400, 101267712, -786948864, -447438528, 248758272, 611981400, 850430336, 1758697920

Offset: 1

N. J. A. Sloane, Nov 15 2004

sign

			G.f. = -24*x - 1472*x^2 - 6048*x^3 + 84480*x^4 - 115920*x^5 - 370944*x^6 + ...
G.f. = -24*q^2 - 1472*q^4 - 6048*q^6 + 84480*q^8 - 115920*q^10 - 370944*q^12 + ...

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

Cf. A000594, A099059.

a[ n_] := SeriesCoefficient[ q QPochhammer[q]^24, {q, 0, 2 n}]; (* Michael Somos, Apr 17 2015 *)
a[n_] := RamanujanTau[2*n]; Array[a, 30] (* Amiram Eldar, Jan 10 2025 *)

{a(n) = if( n<0, 0, n = 2*n; polcoeff( x * eta(x + x * O(x^n))^24, n))}; /* Michael Somos, Apr 17 2015 */

a(n) = A000594(2*n). - Michael Somos, Apr 17 2015

More terms from Joshua Zucker, May 15 2006

A126814 Ramanujan numbers (A000594) read mod 16.

Original entry on oeis.org

1, 8, 12, 0, 14, 0, 8, 0, 5, 0, 4, 0, 6, 0, 8, 0, 2, 8, 12, 0, 0, 0, 8, 0, 7, 0, 8, 0, 6, 0, 0, 0, 0, 0, 0, 0, 14, 0, 8, 0, 10, 0, 4, 0, 6, 0, 0, 0, 9, 8, 8, 0, 14, 0, 8, 0, 0, 0, 4, 0, 6, 0, 8, 0, 4, 0, 12, 0, 0, 0, 8, 0, 10, 0, 4, 0, 0, 0, 0, 0, 9, 0, 12, 0, 12, 0, 8, 0, 10, 0, 0, 0, 0, 0, 8, 0, 2, 8, 4, 0

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. M. Rushforth, Congruence properties of the partition function and associated functions, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 48, No. 3 (1952), pp. 402-413.
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. A000203, A000594.

a[n_] := Mod[RamanujanTau[n], 16]; Array[a, 100] (* Amiram Eldar, Jan 04 2025 *)

a(n) = ramanujantau(n) % 16; \\ Amiram Eldar, Jan 04 2025

a(n) == n^3 * sigma(n) (mod 16) (Rushforth, 1952). - Amiram Eldar, Jan 04 2025

A126816 Ramanujan numbers (A000594) read mod 64.

Original entry on oeis.org

1, 40, 60, 0, 30, 32, 24, 0, 21, 48, 20, 0, 54, 0, 8, 0, 50, 8, 44, 0, 32, 32, 8, 0, 39, 48, 24, 0, 38, 0, 32, 0, 48, 16, 16, 0, 62, 32, 40, 0, 58, 0, 52, 0, 54, 0, 16, 0, 41, 24, 56, 0, 46, 0, 24, 0, 16, 48, 36, 0, 6, 0, 56, 0, 20, 0, 60, 0, 32, 0, 24, 0, 26, 48, 36, 0, 32, 0, 48, 0, 9, 16

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

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.

a[n_] := Mod[RamanujanTau[n], 64]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 64; \\ Amiram Eldar, Jan 05 2025

A126825 Ramanujan numbers (A000594) read mod 3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. P. Bambah and S. Chowla, Congruence properties of Ramanujan’s function tau(n), Bull. Amer. Math. Soc. 53 (1947), 950-955.
John A. Ewell, New representations of Ramanujan's tau function, Proc. Amer. Math. Soc. 128 (2000), 723-726.
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, A000203.

seq(modp(numtheory:-sigma(n),3)*(1-abs(mods(n-1,3))), n=1..105); # Peter Luschny, Apr 26 2016

Mod[RamanujanTau@ #, 3] & /@ Range@ 105 (* Michael De Vlieger, Apr 26 2016 *)

a(n) = ramanujantau(n) % 3; \\ Amiram Eldar, Jan 05 2025

a(4*n) = a(n) (see Corollary 2.2. p. 726 of Ewell link). - Michel Marcus, Dec 23 2012

a(n) = sigma(n) mod 3, for n coprime to 3. - Michel Marcus, Apr 26 2016

A126836 Ramanujan numbers (A000594) read mod 7.

Original entry on oeis.org

1, 4, 0, 5, 0, 0, 0, 4, 2, 0, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 4, 4, 0, 4, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 0, 2, 5, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
R. P. Bambah, Ramanujan’s function tau(n)—A congruence property, Bull. Amer. Math. Soc. 53 (1947), 764-765.
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, A001158.

Cf. this sequence (mod 7^1), A126837 (mod 7^2), A126838 (mod 7^3).

Mod[RamanujanTau@ #, 7] &@ Range@ 120 (* Michael De Vlieger, Apr 25 2016 *)

a(n) = ramanujantau(n) % 7; \\ Amiram Eldar, Jan 05 2025

a(n) = n*sigma_3(n) mod 7. - Michel Marcus, Apr 25 2016

A126837 Ramanujan numbers (A000594) read mod 7^2.

Original entry on oeis.org

1, 25, 7, 47, 28, 28, 14, 4, 37, 14, 22, 35, 21, 7, 0, 31, 28, 43, 0, 42, 0, 11, 46, 28, 32, 35, 28, 21, 23, 0, 0, 31, 7, 14, 0, 24, 46, 0, 0, 14, 14, 0, 37, 5, 7, 23, 42, 21, 0, 16, 0, 7, 36, 14, 28, 7, 0, 36, 42, 0, 14, 0, 28, 7, 0, 28, 15, 42, 28, 0, 2, 1, 14, 23, 28, 0, 14, 0, 39, 35, 46

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Oddmund Kolberg, Note on Ramanujan's Function tau(n), Mathematica Scandinavica, Vol. 10 (1962), pp. 171-172; alternative link.
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, A013957, A126836 (mod 7^1), this sequence (mod 7^2), A126838 (mod 7^3).

a[n_] := Mod[RamanujanTau[n], 49]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 49; \\ Amiram Eldar, Jan 05 2025

a(n) == n * sigma_9(n) (mod 7^2) if Legendre symbol (n,7) = A175629(n) = -1 (Kolberg, 1962). - Amiram Eldar, Jan 05 2025

A126838 Ramanujan numbers (A000594) read mod 7^3.

Original entry on oeis.org

1, 319, 252, 243, 28, 126, 63, 102, 233, 14, 218, 182, 217, 203, 196, 325, 28, 239, 294, 287, 98, 256, 193, 322, 81, 280, 126, 217, 121, 98, 98, 80, 56, 14, 49, 24, 291, 147, 147, 112, 112, 49, 282, 152, 7, 170, 91, 266, 196, 114, 196, 252, 134, 63, 273, 252, 0, 183, 42

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, 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, A126836 (mod 7^1), A126837 (mod 7^2), this sequence (mod 7^3).

a[n_] := Mod[RamanujanTau[n], 343]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 343; \\ Amiram Eldar, Jan 05 2025

A126840 Ramanujan numbers (A000594) read mod 11^2.

Original entry on oeis.org

1, 97, 10, 101, 111, 2, 75, 22, 97, 119, 34, 42, 37, 15, 21, 18, 20, 92, 110, 79, 24, 31, 76, 99, 73, 80, 93, 73, 99, 101, 73, 8, 98, 4, 97, 117, 102, 22, 7, 22, 113, 29, 27, 46, 119, 112, 85, 59, 19, 63, 79, 107, 5, 67, 23, 77, 11, 44, 82, 64, 23, 63, 15, 91, 114, 68, 59, 84, 34

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, 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, A126839 (mod 11^1), this sequence (mod 11^2), A126841 (mod 11^3).

a[n_] := Mod[RamanujanTau[n], 121]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 121; \\ Amiram Eldar, Jan 05 2025

A126841 Ramanujan numbers (A000594) read mod 11^3.

Original entry on oeis.org

1, 1307, 252, 1190, 837, 607, 559, 627, 823, 1208, 881, 405, 1247, 1225, 626, 865, 625, 213, 110, 442, 1113, 152, 1286, 946, 73, 685, 456, 1041, 1067, 948, 194, 855, 1066, 972, 702, 1085, 586, 22, 128, 385, 597, 1239, 269, 893, 724, 1080, 932, 1027, 1229, 910, 442

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, 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, A126839 (mod 11^1), A126840 (mod 11^2), this sequence (mod 11^3).

a[n_] := Mod[RamanujanTau[n], 1331]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 1331; \\ Amiram Eldar, Jan 05 2025

cp's OEIS Frontend

A099059 The odd bisection of A000594.

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A099060 The even bisection of A000594.

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A126814 Ramanujan numbers (A000594) read mod 16.

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A126816 Ramanujan numbers (A000594) read mod 64.

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A126825 Ramanujan numbers (A000594) read mod 3.

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A126836 Ramanujan numbers (A000594) read mod 7.

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A126837 Ramanujan numbers (A000594) read mod 7^2.

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A126838 Ramanujan numbers (A000594) read mod 7^3.

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