|
%I #5 May 11 2023 01:43:37
%S 1,6,8,2,4,4,1,5,1,0,2,3,5,9,3,2,9,3,8,9,5,6,0,0,2,0,3,4,3,1,7,7,1,2,
%T 4,5,3,3,7,2,3,3,6,2,1,3,5,7,9,9,4,9,4,3,8,5,1,5,8,3,5,4,3,9,7,4,9,6,
%U 9,8,9,7,7,6,7,6,0,1,0,6,4,7,8,5,6,2,7,7,7,7,5,4,1,9,7,6,4,3,9,5,5,6,7,5,2
%N Decimal expansion of zeta(3/5) * zeta(4/5) * Product_{p prime} (1 - 1/p^(8/5) - 1/p^(9/5) - 1/p^2 + 1/p^(13/5) + 1/p^(14/5)).
%C The coefficient c_2 of the third term in the asymptotic formula for the number of cubefull numbers (A036966) not exceeding x, N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)) (Bateman and Grosswald, 1958; Finch, 2003).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.6.1, pp. 113-115.
%H Paul T. Bateman and Emil Grosswald, <a href="https://doi.org/10.1215/ijm/1255380836">On a theorem of Erdős and Szekeres</a>, Illinois Journal of Mathematics, Vol. 2, No. 1 (1958), pp. 88-98.
%H P. Shiu, <a href="https://doi.org/10.1017/S0017089500008351">The distribution of cube-full numbers</a>, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295.
%H P. Shiu, <a href="https://doi.org/10.1017/S0305004100070705">Cube-full numbers in short intervals</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 112, No. 1 (1992), pp. 1-5.
%e 1.68244151023593293895600203431771245337233621357994...
%o (PARI) zeta(3/5) * zeta(4/5) * prodeulerrat(1 - 1/p^8 - 1/p^9 - 1/p^10 + 1/p^13 + 1/p^14, 1/5)
%Y Cf. A036966, A090699, A244000, A362973, A362974 (c_0), A362975 (c_1).
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, May 11 2023
|
