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%I #20 Jul 30 2023 02:43:44
%S 3,12,20,28,35,46,52,60,67,76,86,94,101,111,116,124,131,140,148,156,
%T 163,176,182,190,197,206,215,223,230,244,244,252,259,268,276,284,291,
%U 302,308,316,323,332,344,352,359,369,374,382,389,398,406,414,421,433,439,447,454,463,472,480
%N 2-adic valuation of A344014(n).
%C These numbers are integers, even if it should turn out that some of the terms of A344014 are fractional.
%C Jan Vonk observed that a(n) ~ 8*n. - _Robin Visser_, Jul 29 2023
%H Robin Visser, <a href="/A344015/b344015.txt">Table of n, a(n) for n = 1..200</a>
%H Jan Vonk, <a href="https://doi.org/10.1090/bull/1700">Overconvergent modular forms and their explicit arithmetic</a>, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
%F a(n) = A007814(A344014(n)). - _Michel Marcus_, Jul 30 2023
%e From _Robin Visser_, Jul 29 2023: (Start)
%e 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 - ....
%e The first coefficient factors as -744 = -2^3 * 3 * 31, so a(1) = 3.
%e The second coefficient factors as -140914688 = -2^12 * 34403, so a(2) = 12.
%e The third coefficient factors as -16324041375744 = -2^20 * 3 * 79 * 65687, so a(3) = 20. (End)
%o (Sage)
%o def a(n):
%o j1 = sum([1]+[240*sigma(k, 3)*x^k for k in range(1, 2*n)])
%o j2 = product([x]+[(1-x^k)^24 for k in range(1, 2*n)])
%o jinv = (j2/j1^3).taylor(x, 0, 2*n)
%o U2jinv = sum([jinv.coefficient(x^(2*k))*x^k for k in range(0, 2*n)])
%o for k in range(1, n):
%o c = U2jinv.taylor(x, 0, k).coefficient(x^k)
%o U2jinv -= c*(jinv^k)
%o coeff = U2jinv.taylor(x, 0, n).coefficient(x^n)
%o return Integer(coeff).valuation(2) # _Robin Visser_, Jul 29 2023
%Y Cf. A000521, A007814, A344014, A344016, A344017.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jun 17 2021
%E More terms from _Robin Visser_, Jul 29 2023