A000594 -id:A000594 - cp's OEIS frontend

A128381 A007318^24 * A000594.

1, 0, -324, -10976, -260898, -4919184, -67536616, -212659776, 28757879829, 1419855850728, 48139832128404, 1387340166285216, 36039677403855158, 861269016060967824, 18976682736434056968, 379832429643337648960, 6586595998856413482930, 83878202724395340243384, 7117837083803882256428

Offset: 0

Gary W. Adamson, Feb 28 2007

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Conjecture: given the bto performed any k times on A000594 (k = 1, 2, 3, ...); k = 6 and k = 24 are the only members of the set with zeros. k = 6 generates (1, -18, 0, 688, 4494, 5508, ...).

Cf. A007318, A000594, A128378, A128379, A128380, A128382.

Nest[Table[Sum[Binomial[n-1, k-1] * #[[k]], {k, 1, n}], {n, 1, Length[#]}] &, RamanujanTau[Range[19]], 24] (* Amiram Eldar, Jan 08 2025 *)

Binomial transform operation performed 24 times on A000594, assuming A000594 has offset zero.

A278577 Ramanujan function tau(p) as p runs through the prime powers: a(n) = A000594(A000961(n)).

Original entry on oeis.org

1, -24, 252, -1472, 4830, -16744, 84480, -113643, 534612, -577738, 987136, -6905934, 10661420, 18643272, -25499225, -73279080, 128406630, -52843168, -196706304, -182213314, 308120442, -17125708, 2687348496, -1696965207, -1596055698, -5189203740, 6956478662, 2699296768, -15481826884, 9791485272

Offset: 1

N. J. A. Sloane, Nov 29 2016

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Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1933 from Daniel Suteu)
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]

Cf. A000594, A000961, A076847.

Join[{1}, RamanujanTau[Select[Range[100], PrimePowerQ]]] (* Paolo Xausa, May 11 2024 *)

list(lim) = apply(ramanujantau, select(x -> x == 1 || isprimepower(x), vector(lim, i, i))); \\ Amiram Eldar, Jan 09 2025

from itertools import count, islice
from sympy import primefactors, divisor_sigma
def A278577_gen(): # generator of terms
    yield 1
    for n in count(2):
        f = primefactors(n)
        if len(f) == 1:
            p, m = f[0], n+1>>1
            yield (q:=n**4)*(p*n-1)//(p-1)-24*((0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*(m*divisor_sigma(m))**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + q)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m)))
A278577_list = list(islice(A278577_gen(),10)) # Chai Wah Wu, Nov 11 2022

A278578 a(n) = least value of k such that tau(k) = A000594(k) == n mod 23.

Original entry on oeis.org

4, 1, 59, 3481, 5959, 12117361, 351581, 344322938, 995153, 35509681, 1223853461, 117428054, 58714027, 2447706922, 71019362, 1990306, 172161469, 703162, 24234722, 11918, 6962, 118, 2

Offset: 0

N. J. A. Sloane, Nov 29 2016

A finite sequence, containing a(0) through a(22) only.

			tau(4) = -1472 is the first term of A000594 that is a multiple of 23, so a(0) = 4.

Wilton, John Raymond. "Congruence properties of Ramanujan's function τ(n)." Proceedings of the London Mathematical Society 2.1 (1930): 1-10. Table I gives all 23 terms.

J. R. Wilton, Congruence properties of Ramanujan's function τ(n), annotated copy of page 8 only.

Cf. A000594, A106867.

More terms from Ray Chandler, Dec 01 2016

A290152 Coefficients in expansion of E_4*Delta^3 where Delta is the generating function of Ramanujan's tau function (A000594).

Original entry on oeis.org

1, 168, -12636, 392832, -7335174, 92207808, -804651624, 4626614784, -11834988165, -73870961696, 1115908456740, -7498139072256, 32630722986078, -90379426346496, 94395618447768, 450271639673856, -2625847472007243, 6203580643521072, -3151366507609936

Offset: 3

Seiichi Manyama, Jul 21 2017

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Seiichi Manyama, Table of n, a(n) for n = 3..1000
S. C. Milne, Hankel determinants of Eisenstein series, preprint, arXiv:0009130 [math.NT], 2000.

Cf. A000594, A004009 (E_4).

Cf. A290048, A290178.

terms = 19;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*QPochhammer[x]^(3*24) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Let b(q) be the determinant of the 4 X 4 Hankel matrix [E_4, E_6, E_8, E_10 ; E_6, E_8, E_10, E_12 ; E_8, E_10, E_12, E_14 ; E_10, E_12, E_14, E_16]. G.f. is -691^3*3617*b(q)/(1728^3*2^4*3*5^6*7^2*467).

A296993 Numbers k such that k^3 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 24, 32, 64, 96, 128, 256, 288, 384, 512, 1024, 1536, 2048, 4096, 6144, 8192, 16384, 18432, 24576, 32768, 65536, 98304, 131072, 172032, 262144, 276480, 393216, 524288, 1048576, 1179648, 1572864, 1935360, 2097152, 2621440, 3538944, 4194304

Offset: 1

Seiichi Manyama, Dec 22 2017

nonn

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

Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938, A296991, A296992.

from itertools import count, islice
from sympy import divisor_sigma
def A296993_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) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2)))*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) % n**3, count(max(startvalue,1)))
A296993_list = list(islice(A296993_gen(),10)) # Chai Wah Wu, Nov 08 2022

A299204 a(n) = A000594(n) mod (n-1).

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 4, 5, 0, 2, 9, 2, 0, 0, 1, 2, 3, 2, 2, 12, 18, 10, 22, 7, 12, 22, 2, 2, 5, 2, 11, 16, 15, 2, 31, 2, 12, 32, 3, 2, 8, 2, 27, 42, 27, 22, 9, 9, 16, 32, 32, 10, 33, 18, 0, 0, 30, 0, 29, 2, 38, 50, 28, 20, 39, 26, 48, 48, 0, 2, 4, 2, 5, 26, 35, 12

Offset: 2

Seiichi Manyama, Feb 05 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..10000

Cf. A000594, A273650, A299163, A299172, A299205.

f[n_] := Mod[RamanujanTau@n, n - 1]; Array[f, 76, 2] (* Robert G. Wilson v, Feb 07 2018 *)

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

A299205 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, 10, 14, 15, 56, 57, 59, 70, 85, 105, 107, 116, 136, 209, 267, 295, 323, 352, 393, 415, 442, 530, 551, 645, 646, 760, 855, 1197, 1288, 1342, 1415, 1472, 1496, 1625, 1765, 1953, 2002, 2255, 2485, 2847, 2945, 3039, 3382, 3591, 3745, 3905, 4233, 4264, 4313

Offset: 1

Seiichi Manyama, Feb 05 2018

nonn

Numbers k such that A299204(k) = 0.

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

For the sequence when n is prime see A299172.

Cf. A000594, A063938, A299157, A299204.

Select[Range[2, 5000], Divisible[RamanujanTau[#], #-1] &] (* Amiram Eldar, Jan 10 2025 *)

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

A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

Original entry on oeis.org

1, 86, 3750, 109672, 2419462, 43021728, 643548464, 8343640624, 95835049605, 991606081332, 9364586280842, 81571540591968, 661034448807902, 5019357866562208, 35927279225314344, 243657157464337888, 1572638456431119570, 9696997279843999470, 57313953586222481126, 325672739267123628976

Offset: 1

Seiichi Manyama, Sep 11 2018

nonn

			((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(delta^2) =  - 3657830400*q - 314573414400*q^2 - 13716864000000*q^3 - 401161575628800*q^4 - ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics, 185 (3) (2017), 1017-1033.
Wikipedia, Sphere packing

Cf. A000594, A006352 (E_2), A004009 (E_4), A013973 (E_6), A082558, A281373,

About the numerator: A282012 (E_4^4), A282287 (E_6^2*E_4), A282596 (E_6*E_4^2*E_2), A008411 (E_4^3), A280869 (E_6^2), A281374 (E_2^2).

nmax = 25; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); Rest[CoefficientList[Series[-((25*E4[x]^4 - 49*E6[x]^2*E4[x]) + 48*E6[x]*E4[x]^2*E2[x] + (-49*E4[x]^3 + 25*E6[x]^2)* E2[x]^2) / (3657830400 * x^2 * QPochhammer[x]^48), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 12 2018 *)

a(n) ~ exp(4*Pi*sqrt(2*n)) / (132300 * 2^(1/4) * Pi^2 * n^(23/4)). - Vaclav Kotesovec, Sep 12 2018

A034777 Dirichlet convolution of [ 1,1,1,... ] with Ramanujan numbers (A000594).

Original entry on oeis.org

1, -23, 253, -1495, 4831, -5819, -16743, 82985, -113390, -111113, 534613, -378235, -577737, 385089, 1222243, 1070121, -6905933, 2607970, 10661421, -7222345, -4235979, -12296099, 18643273, 20995205, -25494394, 13287951

Offset: 1

N. J. A. Sloane

Multiplicative because both A000012 and A000594 are. - Christian G. Bower, May 16 2005

Inverse Moebius transform of A000594. - Amiram Eldar, Jan 10 2025

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

Cf. A000594.

Table[DivisorSum[k, RamanujanTau], {k, 50}] (* Jan Mangaldan, Mar 15 2013 *)

a(n) = sumdiv(n, d, ramanujantau(d)); \\ Amiram Eldar, Jan 10 2025

A063940 Composite numbers k such that Ramanujan's function tau(k) (A000594) is not divisible by k.

Original entry on oeis.org

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 116, 117, 118, 119, 121, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 143, 145, 146, 148, 152, 153

Offset: 1

Robert G. Wilson v, Aug 31 2001

nonn

			22 is a term because Ramanujan's tau(22) = 18643272 and 18643272 mod 22 = 10.

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

Cf. A000594, A063938, A007659.

Select[ Range[ 70 ], Mod[ CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 70} ] ], 70 ], x ][ [ # ] ], # ] != 0 && ! PrimeQ[ # ] & ]
(* First do *) <Dean Hickerson, Jan 03 2003 *)

cp's OEIS Frontend

A128381 A007318^24 * A000594.

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A278577 Ramanujan function tau(p) as p runs through the prime powers: a(n) = A000594(A000961(n)).

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A278578 a(n) = least value of k such that tau(k) = A000594(k) == n mod 23.

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A290152 Coefficients in expansion of E_4*Delta^3 where Delta is the generating function of Ramanujan's tau function (A000594).

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A296993 Numbers k such that k^3 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

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A299204 a(n) = A000594(n) mod (n-1).

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A299205 Numbers k such that k-1 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

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A319134 Expansion of -((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(3657830400*delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

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Mathematica

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A034777 Dirichlet convolution of [ 1,1,1,... ] with Ramanujan numbers (A000594).

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Author

A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.