A305756 -id:A305756 - cp's OEIS frontend

A305757 Inverse Euler transform of q*(j-720) where j is j-function (A000521).

24, 196584, 16773144, -18919981056, -3292295086056, 2312547886368744, 640457437563740184, -302667453389051314176, -123005476312830648176616, 39529719620247267255853032, 23306082528463942764630528024, -4849033309391159571741461446656

Offset: 1

Seiichi Manyama, Jun 10 2018

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(Conjecture) Let {b_n} = inverse Euler transform of (q*(j+144*k)). b_n is a multiple of 24.

			(1-x)^(-24) * (1-x^2)^(-196584) * (1-x^3)^(-16773144) * (1-x^4)^18919981056 * ... = 1 + 24*x + 196884*x^2 + 21493760*x^3 + 864299970*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..377
N. J. A. Sloane, Transforms

Inverse Euler transform of q*(j+144*k): (-1)*A192731 (k=0), this sequence (k=-5), (-1)*A289061 (k=-12).

Cf. A000521, A007240 (j-720), A302407, A305756.

q*(j-720) = Product_{k>0} (1 - x^k)^(-a(k)).

A305758 Coefficients of (q*(j(q)-720))^(-1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -1, -8191, -690690, 822573570, 142081621501, -102803639194620, -28281091311058949, 13668272700177314310, 5514776417237892113400, -1808663163639294990384630, -1056880191873167904305607180, 224746784451252089897697350159

Offset: 0

Seiichi Manyama, Jun 10 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..378

(q*(j(q)-720))^(m/24): A305760 (m=-24), this sequence (m=-1), A305756 (m=1).

Cf. A000521, A007240 (j(q)-720), A305757.

Convolution inverse of A305756.

A305760 Coefficients of 1/(q*(j(q)-720)) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -24, -196308, -12057152, 38590826190, 5667574866912, -7304962792606024, -1755598494902269440, 1325502689549152990437, 465173370338426065214640, -228213884020015849568089308, -112934890287321570650976240384

Offset: 0

Seiichi Manyama, Jun 10 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..376

(q*(j(q)-720))^(m/24): this sequence (m=-24), A305758 (m=-1), A305756 (m=1).

Cf. A000521, A007240 (j(q)-720), A305699, A305757.

G.f.: Product_{k>0} (1 - x^k)^A305757(k).

A305762 a(0) = 24, a(n) = 2^(max(0, min(3, p - 1))) * 3^(max(0, min(1, q - 1))) where n = 2^p * 3^q * 5^r * ... .

Original entry on oeis.org

24, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 8, 3, 1, 1, 2, 1, 1, 1, 4, 1

Offset: 0

Seiichi Manyama, Jun 10 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A305756.

a[n_] := GCD[24, n/GCD[6, n]]; Array[a, 100, 0] (* Amiram Eldar, Oct 15 2022 *)

a(n)=gcd(24, n/gcd(6,n)) \\ Andrew Howroyd, Jul 24 2018

require 'prime'
def A305762(n)
  return 24 if n == 0
  s = 1
  s *= 3 if n % 9 == 0
  n.prime_division.each{|i|
    s *= 2 ** [3, (i[1] - 1)].min if i[0] == 2
  }
  s
end
p (0..144).map{|i| A305762(i)}

a(n+144) = a(n).
a(n) = gcd(24, n/gcd(6,n)). - Andrew Howroyd, Jul 24 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 77/36. - Amiram Eldar, Oct 15 2022

Keyword:mult added by Andrew Howroyd, Jul 24 2018

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A305757 Inverse Euler transform of q*(j-720) where j is j-function (A000521).

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A305758 Coefficients of (q*(j(q)-720))^(-1/24) where j(q) is the elliptic modular invariant.

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A305760 Coefficients of 1/(q*(j(q)-720)) where j(q) is the elliptic modular invariant.

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A305762 a(0) = 24, a(n) = 2^(max(0, min(3, p - 1))) * 3^(max(0, min(1, q - 1))) where n = 2^p * 3^q * 5^r * ... .

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