A344015 2-adic valuation of A344014(n).

3, 12, 20, 28, 35, 46, 52, 60, 67, 76, 86, 94, 101, 111, 116, 124, 131, 140, 148, 156, 163, 176, 182, 190, 197, 206, 215, 223, 230, 244, 244, 252, 259, 268, 276, 284, 291, 302, 308, 316, 323, 332, 344, 352, 359, 369, 374, 382, 389, 398, 406, 414, 421, 433, 439, 447, 454, 463, 472, 480

Offset

1,1

Comments

These numbers are integers, even if it should turn out that some of the terms of A344014 are fractional.

Jan Vonk observed that a(n) ~ 8*n. - Robin Visser, Jul 29 2023

Links

Robin Visser, Table of n, a(n) for n = 1..200

Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.

Formula

a(n) = A007814(A344014(n)). - Michel Marcus, Jul 30 2023

Example

From Robin Visser, Jul 29 2023: (Start)
An expansion for U_2 j^_1 in terms of powers of j^-1 is given by U_2 j^-1 = -744*j^-1 - 140914688*j^-2 - 16324041375744*j^-3 - ....
The first coefficient factors as -744 = -2^3 * 3 * 31, so a(1) = 3.
The second coefficient factors as -140914688 = -2^12 * 34403, so a(2) = 12.
The third coefficient factors as -16324041375744 = -2^20 * 3 * 79 * 65687, so a(3) = 20. (End)

Prog

(Sage)
def a(n):
    j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 2*n)])
    j2 = product([x]+[(1-x^k)^24 for k in range(1, 2*n)])
    jinv = (j2/j1^3).taylor(x, 0, 2*n)
    U2jinv = sum([jinv.coefficient(x^(2*k))*x^k for k in range(0, 2*n)])
    for k in range(1, n):
        c = U2jinv.taylor(x, 0, k).coefficient(x^k)
        U2jinv -= c*(jinv^k)
    coeff = U2jinv.taylor(x, 0, n).coefficient(x^n)
    return Integer(coeff).valuation(2)  # Robin Visser, Jul 29 2023

Crossrefs

Cf. A000521, A007814, A344014, A344016, A344017.

Sequence in context: A199129 A257942 A063244 * A281813 A063102 A122576

Adjacent sequences: A344012 A344013 A344014 * A344016 A344017 A344018

Keyword

nonn

Author

N. J. A. Sloane, Jun 17 2021

Extensions

More terms from Robin Visser, Jul 29 2023

Status

approved