A000594 -id:A000594 - cp's OEIS frontend

A126812 Ramanujan numbers (A000594) read mod 4.

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

A126833 Ramanujan numbers (A000594) read mod 25.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 6, 5, 7, 5, 12, 6, 12, 6, 10, 11, 16, 7, 20, 15, 12, 12, 22, 10, 0, 12, 20, 18, 5, 10, 7, 21, 24, 16, 5, 21, 11, 20, 24, 0, 17, 12, 17, 11, 10, 22, 21, 22, 18, 0, 7, 11, 2, 20, 10, 5, 15, 5, 10, 5, 12, 7, 17, 18, 10, 24, 16, 23, 19, 5, 22, 10, 22, 11, 0, 10, 22, 24, 5, 5

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, 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George E. Andrews and Bruce C. Berndt, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions, in: Ramanujan's Lost Notebook, Part III, Springer, New York, NY, 2012.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, A013957, A126832 (mod 5^1), this sequence (mod 5^2), A126834 (mod 5^3), A126835 (mod 5^4).

a[n_] := Mod[RamanujanTau[n], 25]; Array[a, 100] (* Amiram Eldar, Jan 04 2025 *)

a(n) = ramanujantau(n) % 25; \\ Amiram Eldar, Jan 04 2025

a(n) == n * sigma_9(n) (mod 25) (Andrews and Berndt, 2012, eq. (5.4.2), p. 98). - Amiram Eldar, Jan 04 2025

A126834 Ramanujan numbers (A000594) read mod 125.

Original entry on oeis.org

1, 101, 2, 28, 80, 77, 6, 105, 107, 80, 112, 56, 12, 106, 35, 11, 66, 57, 45, 115, 12, 62, 22, 85, 25, 87, 45, 43, 5, 35, 82, 71, 99, 41, 105, 121, 61, 45, 24, 25, 67, 87, 42, 11, 60, 97, 121, 22, 43, 25, 7, 86, 52, 45, 85, 5, 90, 5, 10, 105, 37, 32, 17, 18, 85, 124, 116, 98, 44

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Seiichi Manyama, 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.
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, A001157, A013955.

Cf. A126832 (mod 5^1), A126833 (mod 5^2), this sequence (mod 5^3), A126835 (mod 5^4).

Mod[RamanujanTau@ #, 125] & /@ Range@ 69 (* Michael De Vlieger, Apr 26 2016 *)

a(n) = ramanujantau(n) % 125; \\ Amiram Eldar, Jan 05 2025

a(n) = (5*n^2*sigma_7(n) - 4*n*sigma_2(n)) mod 125, for n coprime to 5. - Michel Marcus, Apr 26 2016

A126835 Ramanujan numbers (A000594) read mod 625.

Original entry on oeis.org

1, 601, 252, 403, 455, 202, 131, 105, 107, 330, 237, 306, 387, 606, 285, 261, 316, 557, 170, 240, 512, 562, 147, 210, 150, 87, 295, 293, 380, 35, 582, 571, 349, 541, 230, 621, 436, 295, 24, 275, 442, 212, 542, 511, 560, 222, 371, 147, 418, 150, 257, 336, 552, 420

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, A126832 (mod 5^1), A126833 (mod 5^2), A126834 (mod 5^3), this sequence (mod 5^4).

a[n_] := Mod[RamanujanTau[n], 625]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 625; \\ Amiram Eldar, Jan 05 2025

A126839 Ramanujan numbers (A000594) read mod 11.

Original entry on oeis.org

1, 9, 10, 2, 1, 2, 9, 0, 9, 9, 1, 9, 4, 4, 10, 7, 9, 4, 0, 2, 2, 9, 10, 0, 7, 3, 5, 7, 0, 2, 7, 8, 10, 4, 9, 7, 3, 0, 7, 0, 3, 7, 5, 2, 9, 2, 8, 4, 8, 8, 2, 8, 5, 1, 1, 0, 0, 0, 5, 9, 1, 8, 4, 3, 4, 2, 4, 7, 1, 4, 8, 0, 4, 5, 4, 0, 9, 8, 1, 7, 1, 5, 5, 4, 9, 1, 0, 0, 4, 4, 3, 9, 4, 6, 0, 3, 4, 6, 9, 3, 2, 7, 6

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, this sequence (mod 11^1), A126840 (mod 11^2), A126841 (mod 11^3), A006571.

a[n_] := Mod[RamanujanTau[n], 11]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 11; \\ Amiram Eldar, Jan 05 2025

a(n) = A006571(n) (mod 11), n >= 1. For a proof see the Cowles link under A006571. See also the R. J. Mathar formula there. - Wolfdieter Lang, Feb 16 2016

A144248 Partial sums of A000594.

Original entry on oeis.org

1, -23, 229, -1243, 3587, -2461, -19205, 65275, -48368, -164288, 370324, -620, -578358, -176502, 1040658, 2027794, -4878140, -2150708, 8510712, 1400952, -2818536, -15649224, 2994048, 24283008, -1216217, 12649495, -60629585, -35982417, 92424213, 63212373, 10369205

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 16 2008

sign

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000594.

lst={};s1=0;Do[s1=s1+RamanujanTau[n];AppendTo[lst,s1],{n,5!}];lst
Accumulate[Array[RamanujanTau, 30]] (* Amiram Eldar, Jan 08 2025 *)

a(n)=my(x='x, v=Vec( x * eta(x + x * O(x^(n-1)))^24)); sum(i=1,n,v[i]) \\ Charles R Greathouse IV, Jun 28 2014

A385658 Least prime p < 2n(n+1) such that the polynomial Sum_{k=1..n} tau(k)x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where tau is Ramanujan's tau function given by A000594.

Original entry on oeis.org

1, 2, 5, 17, 59, 19, 43, 17, 19, 89, 47, 67, 257, 89, 173, 11, 103, 67, 103, 191, 29, 89, 101, 139, 19, 13, 19, 79, 79, 271, 223, 149, 131, 5, 37, 31, 593, 149, 353, 109, 293, 293, 17, 19, 97, 83, 59, 79, 883, 101, 71, 13, 199, 113, 1013, 29, 1279, 7, 181, 383, 269, 197, 17

Offset: 1

Zhi-Wei Sun, Aug 03 2025

nonn

Conjecture: a(n) > 1 for all n > 1. In other words, for each n = 2,3,... the polynomial x^(n-1) + tau(2)*x^(n-2) + ... + tau(n) is irreducible modulo some prime p < 2n*(n+1).

			a(5) = 59 since the prime 59 is smaller than 2*5*(5+1) = 60, and 59 is the least prime p such that the polynomial tau(1)*x^4 + tau(2)*x^3 + tau(3)*x^2 + tau(4)*x + tau(5) is irreducible modulo p.

Zhi-Wei Sun, Table of n, a(n) for n = 1..500

Cf. A000040, A000594, A385676, A385678.

Tau[n_]:=Tau[n]=RamanujanTau[n];
P[n_,x_]:=P[n,x]=Sum[Tau[k]x^(n-k),{k,1,n}];
tab={};Do[Do[If[IrreduciblePolynomialQ[P[n, x], Modulus->Prime[k]]==True, tab=Append[tab,Prime[k]]; Goto[aa]], {k, 1, PrimePi[2n(n+1)-1]}];tab=Append[tab,1]; Label[aa]; Continue, {n,1,63}];Print[tab]

a(n) = forprime(p=2, 2*n*(n+1)-1, if (polisirreducible(Mod(sum(k=1, n, ramanujantau(k)*x^(n-k)), p)), return(p))); 1; \\ Michel Marcus, Aug 04 2025

A034778 Dirichlet convolution of Ramanujan numbers (A000594) with themselves.

Original entry on oeis.org

1, -48, 504, -2368, 9660, -24192, -33488, 239616, -163782, -463680, 1069224, -1193472, -1155476, 1607424, 4868640, 86016, -13811868, 7861536, 21322840, -22874880, -16877952, -51322752, 37286544, 120766464, -27669550, 55462848, -203834232

Offset: 1

N. J. A. Sloane

Multiplicative because A000594 is. - Christian G. Bower, May 16 2005

			G.f. = x - 48*x^2 + 504*x^3 - 2368*x^4 + 9660*x^5 - 24192*x^6 - 33488*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000594.

a[n_] := DivisorSum[n, RamanujanTau[#]*RamanujanTau[n/#]&]; Array[a, 30] (* Jean-François Alcover, Nov 14 2015 *)

{a(n) = local(A); if( n<1, 0, A = Vec( eta(x + x^n*O(x))^24); sumdiv(n, d, A[d] * A[n/d]))}; /* Michael Somos, Jul 16 2004 */

use ntheory ":all"; for my $n (1..50) { say divisor_sum($n, sub { my $d=shift; ramanujan_tau($d)*ramanujan_tau($n/$d) } # Dana Jacobsen, Sep 05 2015

a(n) = Sum_{d|n} tau(d)tau(n/d) where tau(n) = A000594(n) is Ramanujan's tau function.

A035174 Ramanujan's tau function (or tau numbers (A000594)) for 2^n.

Original entry on oeis.org

1, -24, -1472, 84480, 987136, -196706304, 2699296768, 338071388160, -13641873096704, -364965248630784, 36697722069188608, -133296500464680960, -71957818786545926144, 1999978883828768833536, 99370119662955604738048

Offset: 0

Robert G. Wilson v, Jan 04 2003

Seiichi Manyama, Table of n, a(n) for n = 0..604
Index entries for linear recurrences with constant coefficients, signature (-24,-2048).

Cf. A000594, A064556.

Mathematica
```
Table[ RamanujanTau[2^n], {n, 0, 14}]
```

a(n)=sum(j=0,n\2,(-1)^j*binomial(n-j, n-2*j)*2^(11*j)*(-24)^(n-2*j)) \\ Charles R Greathouse IV, Apr 28 2013

Vec(1/(1+24*x+2048*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 28 2013

use ntheory ":all"; say "$ ",ramanujan_tau(1 << $) for 0..63; # Dana Jacobsen, Sep 05 2015

G.f.: 1/(1 + 24x + 2048x^2). Proof by Robin Chapman: Follows from the formula tau(p^{n+2}) = tau(p)tau(p^{n+1}) - p^11 tau(p^n) for prime p, which comes from the theory of Hecke operators on modular forms. The p = 2 case gives a recurrence for tau(2^n) leading immediately to the g.f.

A064556 Ramanujan's tau function (or tau numbers (A000594)) for 10^n.

Original entry on oeis.org

1, -115920, 37534859200, -30328412970240000, -482606811957501440000, -2983637890141033828147200000

Offset: 0

Robert G. Wilson v, Jan 04 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..181
Index entries for linear recurrences with constant coefficients, signature (-115920, 124097412800, -11592000000000000, -10000000000000000000000).

Cf. A000594, A035174.

Mathematica
```
Table[ RamanujanTau[10^n], {n, 0, 5}]
```

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)=taup(5,n)*taup(2,n) \\ Charles R Greathouse IV, Sep 06 2015

use bigint; use ntheory ":all"; say "$ ",ramanujan_tau(10 ** $) for 0..19; # Dana Jacobsen, Sep 05 2015

cp's OEIS Frontend

A126812 Ramanujan numbers (A000594) read mod 4.

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A126833 Ramanujan numbers (A000594) read mod 25.

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A126834 Ramanujan numbers (A000594) read mod 125.

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A126835 Ramanujan numbers (A000594) read mod 625.

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A126839 Ramanujan numbers (A000594) read mod 11.

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A144248 Partial sums of A000594.

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A385658 Least prime p < 2n*(n+1) such that the polynomial Sum_{k=1..n} tau(k)*x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where tau is Ramanujan's tau function given by A000594.

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A034778 Dirichlet convolution of Ramanujan numbers (A000594) with themselves.

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A385658 Least prime p < 2n(n+1) such that the polynomial Sum_{k=1..n} tau(k)x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where tau is Ramanujan's tau function given by A000594.