A000594 -id:A000594 - cp's OEIS frontend

A063938 Numbers k that divide tau(k), where tau(k)=A000594(k) is Ramanujan's tau function.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 88, 90, 91, 92, 96, 98, 100, 105, 108, 112, 115, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 161, 162, 168

Offset: 1

Robert G. Wilson v, Aug 31 2001

Although most small numbers are in the sequence, it becomes sparser for larger values; e.g., only 504 numbers up to 10000 and only 184 numbers from 10001 to 20000 are in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Tau Function.

For the sequence when n is prime see A007659.

Cf. A063940, A000594, A079334, A296991, A296993.

(* First do <Michael De Vlieger, Dec 23 2017 *)

for (n=1,1000,if(Mod(ramanujantau(n),n)==0,print1(n", "))) \\ Dana Jacobsen, Sep 06 2015

use ntheory ":all"; my @p = grep { !(ramanujan_tau($) % $) } 1..1000; say "@p"; # Dana Jacobsen, Sep 06 2015

from itertools import count, islice
from sympy import divisor_sigma
def A063938_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not -840*(pow(m:=n+1>>1,2,n)*(0 if n&1 else pow(m*divisor_sigma(m),2,n))+(sum(pow(i,4,n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))<<1)) % n, count(max(startvalue,1)))
A063938_list = list(islice(A063938_gen(),25)) # Chai Wah Wu, Nov 08 2022

More terms from Dean Hickerson, Jan 03 2003

A027860 a(n) = (-tau(n) + sigma_11(n)) / 691, where tau is Ramanujan's tau (A000594), sigma_11(n) = Sum_{ d divides n } d^11 (A013959).

Original entry on oeis.org

0, 3, 256, 6075, 70656, 525300, 2861568, 12437115, 45414400, 144788634, 412896000, 1075797268, 2593575936, 5863302600, 12517805568, 25471460475, 49597544448, 93053764671, 168582124800, 296526859818, 506916761600, 846025507836, 1378885295616, 2203231674900

Offset: 1

Bill Gosper

nonn

It appears that this sequence is strictly increasing. - Jianing Song, Aug 05 2018

"Number Theory I", vol. 49 of the Encyc. of Math. Sci.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000594, A013959.

Similar sequences: A281788, A281876, A281928, A281956, A281979.

(sum(n^11*q^n/(1-q^n), n,1,inf)-q*prod(1-q^n,n,1,inf)^24)/691; taylor(%,q,0,24);

N:= 100: # to get a(1) to a(N)
S:= series(q*mul((1-q^k)^24,k=1..N),q,N+1):
seq((-coeff(S,q,n) + add(d^11, d = numtheory:-divisors(n)))/691, n=1..N); # Robert Israel, Nov 12 2014

{0}~Join~Array[(-RamanujanTau@ # + DivisorSigma[11, #])/691 &, 24] (* Michael De Vlieger, Aug 05 2018 *)

a(n) = (sigma(n, 11) - polcoeff( x * eta(x + x * O(x^n))^24, n))/691; \\ for n>0; Michel Marcus, Nov 12 2014

def A027860List(len):
    r = list(delta_qexp(len+1))
    return [(sigma(n, 11) - r[n])/691 for n in (1..len)]
A027860List(24) # Peter Luschny, Aug 20 2018

a(n) = (A013959(n) - A000594(n))/691. - Michel Marcus, Nov 12 2014

More terms from Michel Marcus, Nov 12 2014

A121733 Numbers k such that tau(k) = tau(k+1) mod 691, where tau is Ramanujan's tau function A000594.

Original entry on oeis.org

184, 2103, 3421, 3638, 4342, 5181, 6029, 6233, 8323, 8628, 8721, 9658, 9905, 11322, 11774, 11888, 12410, 12774, 12811, 13063, 13484, 14744, 14906, 15065, 15247, 16581, 16610, 18248, 18396, 18703, 19514, 20476, 20479, 21657, 22089, 22984

Offset: 1

Alexander Adamchuk, Aug 18 2006

nonn

Corresponding Ramanujan tau numbers mod 691 are listed in A121734(n) = A046694(a(n)). A121734 begins 483, 209, 21, 632, 650, 541, 546, 281, 666, 440, 397, 576, 18, 251, 356, 207, 532, 361, 121, 642, 288, 167, 348, 505, 561, 0, 108, 166, 97, 492, 58, 255, 632, 151, 679, 185, 141, 587, 0, ....

There are instances of three consecutive equal terms in A046694, with A046694(n) = A046694(n+1) = A046694(n+2). Equivalently there are consecutive equal terms a(n) = a(n+1). The first is A046694(290217) = A046694(290218) = A046694(290219) = 0. - Alexander Adamchuk, Aug 18 2006

			a(1) = 184 because the first pair of equal consecutive numbers in A046694 is A046694(184) = A046694(185) = 483 = A121734(1).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Charles R Greathouse IV)
Eric Weisstein's World of Mathematics, Ramanujan's Tau Function.

Cf. A000594, A046694, A121734.

Select[Range[30000],Mod[DivisorSigma[11,#1],691]==Mod[DivisorSigma[11,#1+1],691]&]

is(n)=(ramanujantau(n)-ramanujantau(n+1))%691==0 \\ Charles R Greathouse IV, Feb 08 2017

A121734 Ramanujan tau numbers such that A000594(k) == A000594(k+1) mod 691, or A046694(k) = A046694(k+1).

Original entry on oeis.org

483, 209, 21, 632, 650, 541, 546, 281, 666, 440, 397, 576, 18, 251, 356, 207, 532, 361, 121, 642, 288, 167, 348, 505, 561, 0, 108, 166, 97, 492, 58, 255, 632, 151, 679, 185, 141, 587, 0, 549, 459, 428, 549, 157, 559, 121, 605, 102

Offset: 1

Alexander Adamchuk, Aug 18 2006

nonn

The corresponding indices k are listed in A121733.

			a(1) = 483 because the first pair of equal consecutive numbers in A046694 is A046694(184) = A046694(185) = 483.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ramanujan's Tau Function.

Cf. A000594, A046694, A121733.

Do[f=Mod[DivisorSigma[11,n],691];g=Mod[DivisorSigma[11,n+1],691];If[f==g,Print[{n,f}]],{n,1,10000}]

a(n) = mod(A000594(A121733(n)), 691) = A046694(A121733(n)).

A121742 Numbers k such that three consecutive Ramanujan tau numbers are congruent mod 691, or A000594(k) == A000594(k+1) == A000594(k+2) mod 691, or A046694(k) = A046694(k+1) = A046694(k+2).

Original entry on oeis.org

290217, 477155, 1051085, 1153412, 1409635, 1409636, 1641812, 2056412, 2657865, 2945116, 3724928, 4570784, 5115359, 5187777, 5567783, 5720418, 7836078, 8736807, 8932428, 9618716, 9957630, 10175867, 10447914, 10547421, 10982172, 11359120, 11499876, 11735611, 12651355, 13018169, 13515452, 13867914

Offset: 1

Alexander Adamchuk, Aug 19 2006

nonn

Corresponding Ramanujan tau numbers mod 691 are listed in A121743(n) = A046694(a(n)). A121743(n) begins {0,276,91,79,0,0,...}. a(n) are the indices of the first number in the Ramanujan tau triples mod 691. All a(n) belong to A121733(n) - indices of the first number in the Ramanujan tau twins mod 691. There are also quadruplets in the Ramanujan tau mod 691 such that A046694(n) = A046694(n+1) = A046694(n+2) = A046694(n+3). The first such Ramanujan tau quadruplet mod 691 starts with A046694(1409635) = 0.

Jud McCranie, Table of n, a(n) for n = 1..2568
Eric Weisstein Ramanujan's Tau Function.

Cf. A000594, A046694, A121733, A121734, A121743.

Do[f=Mod[DivisorSigma[11,n],691];g=Mod[DivisorSigma[11,n+1],691];h=Mod[DivisorSigma[11,n+2],691];If[f==g&&g==h,Print[{n,f}]],{n,1,1500000}]

a(7)-a(16) from Amiram Eldar, Jan 26 2020

More terms from Jud McCranie, Nov 02 2020

A126832 Ramanujan numbers (A000594) read mod 5.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 166-167.

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. A000203, A000594, A064987.

Cf. this sequence (mod 5^1), A126833 (mod 5^2), A126834 (mod 5^3), A126835 (mod 5^4).

Mod[RamanujanTau@ #, 5] & /@ Range@ 105 (* Michael De Vlieger, Apr 26 2016 *)

a(n) = n*sigma(n) % 5; \\ Amiram Eldar, Jan 05 2025

from sympy import divisor_sigma
def A126832(n): return n*divisor_sigma(n)%5 # Chai Wah Wu, Aug 24 2023

a(n) = n*sigma(n) mod 5. - Michel Marcus, Apr 26 2016. See also the Hardy reference, p. 166, (10.5.2), with a proof. - Wolfdieter Lang, Feb 03 2017

A273650 a(n) = A000594(n) mod n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 10, 0, 7, 0, 0, 20, 1, 0, 0, 16, 0, 0, 24, 0, 21, 0, 21, 32, 0, 0, 31, 22, 27, 0, 30, 0, 31, 24, 0, 22, 27, 0, 0, 0, 21, 28, 29, 0, 45, 0, 54, 4, 14, 0, 49, 54, 0, 0, 30, 24, 64, 36, 45, 0, 19, 0, 67, 70, 0, 32, 42, 54, 37, 0, 0, 18

Offset: 1

Seiichi Manyama, May 27 2016

nonn

			tau(10) mod 10 = (-115920) mod 10 = 0,
tau(11) mod 11 = 534612 mod 11 = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000594, A063938, A295654.

a[n_] := Mod[RamanujanTau[n], n]; Array[a, 100] (* Amiram Eldar, Jan 08 2025 *)

a(n)=ramanujantau(n)%n \\ assumes the GRH; Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A273650(n): return -840*(pow(m:=n+1>>1,2,n)*(0 if n&1 else pow(m*divisor_sigma(m),2,n))+(sum(pow(i,4,n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))<<1)) % n # Chai Wah Wu, Nov 08 2022

a(n) = A000594(n) mod n.
From Amiram Eldar, Jan 08 2025: (Start)
a(A063938(n)) = 0.
abs(a(A295654(n))) = 1. (End)

A290048 Coefficients in expansion of E_6*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -552, 8640, 116000, -4868460, 67855536, -544522240, 2742137280, -8237774250, 10592091400, 3366617856, 113971542048, -1217020425880, 4535746506000, -5415752171520, -19090509870144, 93580817811453, -142801363479240, -80721277168000, 665065363025280

Offset: 2

Seiichi Manyama, Jul 19 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

Cf. A000594, A010839, A013973 (E_6).
Cf. A282382, A282461 (E_6*E_10*E_14 = E_10^3), A290049, A290050.
E_k*Delta^2: A290178 (k=4), this sequence (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).

terms = 20;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_6, E_8, E_10 ; E_8, E_10, E_12 ; E_10, E_12, E_14]. G.f. is -691^2*b(q)/(1728^2*250^2).

a(n) = (A290050(n) - 2*691*A290049(n) + 691^2*A282382(n))/(1728^2*250^2).

A290178 Coefficients in expansion of E_4*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 192, -8280, 147200, -1438020, 7491456, -4626880, -246965760, 2112385950, -9443825600, 23625035616, -14413771008, -118710609640, 427914230400, -467038103040, -645319017984, 1640006523477, 2800373100480, -8506579320400, -21655683517440, 108181106829972

Offset: 2

Seiichi Manyama, Jul 23 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 2..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

E_k*Delta^2: this sequence (k=4), A290048 (k=6), A290180 (k=8), A290181 (k=10), A290182 (k=14).

Cf. A000594, A004009 (E_4), A290152.

terms = 21;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 3 X 3 matrix [E_4, E_8, E_10 ; E_6, E_10, E_12 ; E_8, E_12, E_14]. G.f. is 691^2*b(q)/(1728^2*21^2*250).

A128379 A000012^23 * A000594.

Original entry on oeis.org

1, -1, -24, 0, 276, 300, -1748, -4300, 4278, 29026, 22724, -94668, -242398, -18722, 856980, 1472252, -384491, -5299269, -7824968, 2088032, 25655442, 38814478, -69160, -99735912, -175711283, -68736397, 294769680, 686373176, 562588924, -513324396, -2155273788, -2808874356

Offset: 1

Gary W. Adamson, Feb 28 2007

sign

Conjecture: given A000012^k * A000594, k=23 and 24 are the only k's generating sequences with zeros. k = 24 in A128378: (1, 0, -24, -24, 252, 552, -1196, -5496, ...).

Cf. A000594, A128378, A144248, A144249.

Nest[Accumulate, RamanujanTau[Range[32]], 23] (* Amiram Eldar, Jan 08 2025 *)

A000012 (partial sum operator) performed 23 times on A000594.

cp's OEIS Frontend

A063938 Numbers k that divide tau(k), where tau(k)=A000594(k) is Ramanujan's tau function.

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A027860 a(n) = (-tau(n) + sigma_11(n)) / 691, where tau is Ramanujan's tau (A000594), sigma_11(n) = Sum_{ d divides n } d^11 (A013959).

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A121733 Numbers k such that tau(k) = tau(k+1) mod 691, where tau is Ramanujan's tau function A000594.

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A121734 Ramanujan tau numbers such that A000594(k) == A000594(k+1) mod 691, or A046694(k) = A046694(k+1).

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A121742 Numbers k such that three consecutive Ramanujan tau numbers are congruent mod 691, or A000594(k) == A000594(k+1) == A000594(k+2) mod 691, or A046694(k) = A046694(k+1) = A046694(k+2).

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A126832 Ramanujan numbers (A000594) read mod 5.

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A273650 a(n) = A000594(n) mod n.

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