A296993 -id:A296993 - 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

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

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

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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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A296991 Numbers k such that k^2 divides 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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Mathematica

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