A000594 -id:A000594 - cp's OEIS frontend

A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

1, 432, 39960, -1418560, 17312940, -71928864, -462815680, 7500885120, -38038437810, 29000909200, 729783353376, -4661016429888, 13691625085880, -16503845217120, -14982974507520, 45085348093056, 99234456545637, -157805792764560, -1644659689877680

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: A290178 (k=4), A290048 (k=6), this sequence (k=8), A290181 (k=10), A290182 (k=14).

Cf. A000594, A008410 (E_8).

terms = 19;
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
E8[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_6, E_8, E_12 ; E_8, E_10, E_14 ; E_10, E_12, E_16]. G.f. is -691^2*3617*b(q)/(1728^2*2^3*3*5^3*7^2*467).

A290181 Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -312, -121680, 1004000, 37942020, -801594864, 6139193600, -11831002560, -151614128250, 1346611783000, -4592794000704, 3738595861728, 15192491492360, 47281379454000, -737660590018560, 2662090686805056, -3290770281735027, -4884703150768920

Offset: 2

Seiichi Manyama, Jul 23 2017

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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: A290178 (k=4), A290048 (k=6), A290180 (k=8), this sequence (k=10), A290182 (k=14).

Cf. A000594, A013974 (E_10).

terms = 18;
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
E10[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_6, E_10, E_12 ; E_8, E_12, E_14 ; E_10, E_14, E_16]. G.f. is 691^2*3617*b(q)/(1728^2*2^2*3*5^6*7^2*13).

A290182 Coefficients in expansion of E_14*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, -72, -194400, -28866400, 13994100, 9650004336, -99683138560, -1007380800, 5570606272950, -32186306471000, -2717893793664, 724443400725408, -2662202398202200, -401005712372400, 19385312101171200, 24633489938571456, -449375771787124707

Offset: 2

Seiichi Manyama, Jul 23 2017

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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: A290178 (k=4), A290048 (k=6), A290180 (k=8), A290181 (k=10), this sequence (k=14).

Cf. A000594, A058550 (E_14).

terms = 17;
E14[x_] = 1 - 24*Sum[k^13*x^k/(1 - x^k), {k, 1, terms}];
E14[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_8, E_10, E_14 ; E_10, E_12, E_16 ; E_12, E_14, E_18]. G.f. is -691^2*3617*43867*b(q)/(1728^2*2^6*3*5^3*7^2*97*7213).

A296991 Numbers k such that k^2 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 21, 24, 27, 32, 36, 40, 42, 48, 54, 64, 72, 81, 84, 96, 108, 120, 128, 135, 144, 162, 168, 189, 192, 216, 243, 256, 270, 280, 288, 324, 336, 360, 378, 384, 432, 448, 486, 512, 540, 576, 640, 648, 672, 729, 756, 768, 828, 840, 864

Offset: 1

Seiichi Manyama, Dec 22 2017

nonn

2^k is a term for k >= 0.

Seiichi Manyama, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Tau Function

Cf. A000594, A063938, A296992, A296993.

fQ[n_] := Mod[RamanujanTau@n, n^2] == 0; Select[Range@875, fQ] (* Robert G. Wilson v, Dec 23 2017 *)

is(n) = Mod(ramanujantau(n), n^2)==0 \\ Felix Fröhlich, Dec 24 2017

from itertools import count, islice
from sympy import divisor_sigma
def A296991_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not -24*((m:=n+1>>1)**2*(0 if n&1 else m*(35*m - 52*n)*divisor_sigma(m)**2)+sum(i**3*(70*i - 140*n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) % n**2, count(max(startvalue,1)))
A296991_list = list(islice(A296991_gen(),20)) # Chai Wah Wu, Nov 08 2022

A070563 a(n) = 0 if 3 divides the Ramanujan number tau(n) (A000594(n)), otherwise 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 07 2002

Multiplicative because A000594 is. Conjecture: a(3^k) = 0, if p == 1 mod 3, a(p^2k) = 0 and a(p^(2k+1)) = 1, if p == -1 mod 3, a(p^2k) = 1 and a(p^(2k+1)) = 0. - Christian G. Bower, Jun 10 2005
From Antti Karttunen, Jul 03 2024: (Start)
The above conjecture is not correct. The first counterexample occurs at n = 2401 = 7^4. My improved conjecture is that this is actually a characteristic function of nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3, that is, having a following multiplicative formula: a(3^k) = 0, if p == 1 mod 3, a(p^e) = 1 if e != 2 (mod 3), otherwise 0, and if p == -1 mod 3, a(p^2k) = 1 and a(p^(2k+1)) = 0. This conjecture has now been proved correct by Seiichi Manyama.
Bower's formula is now submitted as A374053.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..131072 (first 100000 terms from Antti Karttunen)
P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.
P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.
Index entries for characteristic functions

Characteristic function of A374135, nonmultiples of 3 whose sum of divisors is also a nonmultiple of 3.

Cf. A000203, A000594, A011655, A070564, A126825, A353815, A374053.

a[n_] := Boole[!Divisible[RamanujanTau[n], 3]]; Array[a, 92] (* Jean-François Alcover, Jul 05 2017 *)

A070563(n) = !!(ramanujantau(n)%3); \\ Antti Karttunen, Jul 02 2024

A070563(n) = ((n%3) && (sigma(n)%3)); \\ Antti Karttunen, Jul 03 2024

A070563(n) = { my(f=factor(n)); prod(i=1, #f~, if(3==f[i, 1], 0, 1==(f[i, 1]%3), 2!=(f[i, 2]%3), (1+f[i, 2])%2)); }; \\ Antti Karttunen, Jul 03 2024

a(n) = A011655(n) * A353815(n), conjectured by Antti Karttunen, proved by Seiichi Manyama, Jul 03 2024

A128380 A097806^24 * A000594.

Original entry on oeis.org

1, 0, -48, -24, 1104, 1128, -15892, -25368, 156240, 360640, -1057908, -3600696, 4417678, 26438568, -3155508, -144207816, -112109568, 564538680, 1002957320, -1344487080, -5096138658, -111333800, 17182425012, 17552839368, -34668142443, -86942440944, 4993723500, 236551774320

Offset: 1

Gary W. Adamson, Feb 28 2007

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Conjecture: Given the infinite set of sequences generated from the pairwise operation on A000594 (A097806^k * A000594), k = 24, (A128380) is the only sequence in the set with a zero. The sequence generated from k=23 = (1, -1, -47, 23, 1081, 47, -15939, ...). Analogous conjecture with the partial sum operator: (Cf. A128378, A128379); in which zeros are conjectured to occur only with k=23 and k=24. A128380 mod 24 = 1, 0, 0, 0, 0, 0, -4, 0, 0, 16, ...

Cf. A000594, A097806, A128378, A128379.

Nest[Prepend[Most[#] + Rest[#], First[#]] &, RamanujanTau[Range[30]], 24] (* Amiram Eldar, Jan 08 2025 *)

Pairwise operation performed 24 times on A000594

More terms from Amiram Eldar, Jan 08 2025

A295645 Primes p such that tau(p) +- 1 is congruent to 0 (mod p), where tau is the Ramanujan tau function (A000594).

Original entry on oeis.org

11, 23, 691, 5807

Offset: 1

Seiichi Manyama, Nov 25 2017

Nik Lygeros and Olivier Rozier found a new solution to the equation tau(p) + 1 == 0 (mod p) for prime p = 692881373, on September 6 2009. - Seiichi Manyama, Dec 30 2017
a(5) > 8*10^7. - Seiichi Manyama, Jan 01 2018
A superset of A193855. - Jud McCranie, Nov 06 2020

			tau(11) = 534612 and 11 | (534612 - 1), so a(1) = 11.
tau(23) = 18643272 and 23 | (18643272 - 1), so a(2) = 23.
tau(691) = -2747313442193908 and 691 | (-2747313442193908 - 1), so a(3) = 691.
tau(5807) = 237456233554906855056 and 5807 | (237456233554906855056 + 1), so a(4) = 5807.

N. Lygeros and O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.
Wikipedia, Ramanujan tau function.

Cf. A000594, A076847 (tau(p)), A007659, A193855, A273650, A295654.

Select[Prime@ Range[10^3], Function[p, AnyTrue[RamanujanTau[p] + {-1, 1}, Divisible[#, p] &]]] (* Michael De Vlieger, Dec 30 2017 *)

isok(p) = my(rp=ramanujantau(p)); isprime(p) && !((rp-1) % p) || !((rp+1) % p); \\ Michel Marcus, Nov 07 2020

A296992 Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.

Original entry on oeis.org

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

Offset: 2

Seiichi Manyama, Dec 22 2017

nonn

			tau(2) =   -24 and 2^3 divides   24, so a(2) = 3.
tau(3) =   252 and 3^2 divides  252, so a(3) = 2.
tau(4) = -1472 and 4^3 divides 1472, so a(4) = 3.

Amiram Eldar, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A063938 (a(n)>=1), A296991 (a(n)>=2), A296993 (a(n)>=3).

Cf. A191599 (a(n)=0), A297000 (a(n)=1), A297001 (a(n)=2).

f[n_] := Block[{m = 0}, While[Mod[RamanujanTau@n, n^m] == 0, m++]; m - 1]; Array[f, 93, 2] (* Robert G. Wilson v, Dec 23 2017 *)
a[n_] := IntegerExponent[RamanujanTau[n], n]; Array[a, 100, 2] (* Amiram Eldar, Jan 09 2025 *)

a(n) = valuation(ramanujantau(n), n); \\ Amiram Eldar, Jan 09 2025

A299157 Numbers k such that k+1 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 13, 17, 19, 20, 22, 23, 27, 29, 31, 41, 45, 47, 53, 55, 59, 68, 71, 76, 77, 79, 83, 87, 89, 91, 97, 104, 107, 114, 127, 137, 139, 149, 160, 167, 171, 177, 179, 183, 191, 195, 199, 209, 223, 229, 239, 240, 243, 251, 269, 275, 293, 297, 321, 343

Offset: 1

Seiichi Manyama, Feb 04 2018

nonn

Numbers k such that A299163(k) = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938, A079334, A296991, A299158, A299163.

q[k_] := Divisible[RamanujanTau[k], k+1]; Select[Range[350], q] (* Amiram Eldar, Jan 08 2025 *)

isok(n) = (ramanujantau(n) % (n+1)) == 0; \\ Michel Marcus, Feb 05 2018

A299163 a(n) = A000594(n) mod (n+1).

Original entry on oeis.org

1, 0, 0, 3, 0, 0, 0, 6, 7, 9, 0, 11, 0, 6, 8, 14, 0, 1, 0, 0, 2, 0, 0, 10, 15, 24, 0, 10, 0, 18, 0, 30, 12, 21, 12, 20, 30, 6, 24, 4, 0, 3, 16, 21, 0, 15, 0, 21, 43, 15, 20, 21, 0, 45, 0, 27, 42, 34, 0, 28, 46, 42, 56, 38, 48, 60, 16, 0, 14, 63, 0, 50, 60, 36, 12

Offset: 1

Seiichi Manyama, Feb 04 2018

nonn

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

Cf. A000594, A273650, A299157.

a[n_] := Mod[RamanujanTau[n], n+1]; Array[a, 100] (* Amiram Eldar, Jan 09 2025 *)

PARI
```
{a(n) = ramanujantau(n)%(n+1)}
```

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A290180 Coefficients in expansion of E_8*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A290181 Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A290182 Coefficients in expansion of E_14*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

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A296991 Numbers k such that k^2 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

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A070563 a(n) = 0 if 3 divides the Ramanujan number tau(n) (A000594(n)), otherwise 1.

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A128380 A097806^24 * A000594.

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A295645 Primes p such that tau(p) +- 1 is congruent to 0 (mod p), where tau is the Ramanujan tau function (A000594).

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A296992 Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.

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