A296992 -id:A296992 - cp's OEIS frontend

A297000 Numbers k such that A296992(k) = 1.

5, 7, 10, 14, 15, 20, 25, 28, 30, 35, 45, 49, 50, 56, 60, 63, 70, 75, 80, 88, 90, 91, 92, 98, 100, 105, 112, 115, 125, 126, 140, 147, 150, 160, 161, 175, 180, 182, 196, 200, 207, 210, 224, 225, 230, 240, 245, 250, 252, 264, 273, 276, 294, 300, 315, 320, 322, 343

Offset: 1

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

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938, A296992, A297001.

1 + Position[Table[-1 + SelectFirst[Range[0, 5], ! Divisible[RamanujanTau@ n, n^#] &], {n, 2, 350}], 1][[All, 1]] (* Michael De Vlieger, Dec 23 2017 *)

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

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

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

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

cp's OEIS Frontend

A297000 Numbers k such that A296992(k) = 1.

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

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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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Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python