A296991 -id:A296991 - 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

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

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

A297001 Numbers k such that A296992(k) = 2.

Original entry on oeis.org

3, 9, 12, 18, 21, 27, 36, 40, 42, 48, 54, 72, 81, 84, 108, 120, 135, 144, 162, 168, 189, 192, 216, 243, 270, 280, 324, 336, 360, 378, 432, 448, 486, 540, 576, 640, 648, 672, 729, 756, 768, 828, 840, 864, 945, 972, 1080, 1152, 1296, 1344, 1458, 1512, 1701, 1728, 1890

Offset: 1

Seiichi Manyama and Robert G. Wilson v, Dec 23 2017

nonn

Amiram Eldar, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A296991, A296992, A297000.

1 + Position[Table[-1 + SelectFirst[Range[0, 5], !Divisible[RamanujanTau@ n, n^#] &], {n, 2, 2000}], 2][[All, 1]] (* Michael De Vlieger, Dec 23 2017 *)

isok(k) = if(k == 1, 0, valuation(ramanujantau(k), k) == 2); \\ Amiram Eldar, Jan 09 2025

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

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

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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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A297001 Numbers k such that A296992(k) = 2.

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