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A362973 The number of cubefull numbers (A036966) not exceeding 10^n.

%I A362973 #35 Jan 01 2024 11:56:28
%S A362973 1,2,7,20,51,129,307,713,1645,3721,8348,18589,41136,90619,198767,
%T A362973 434572,947753,2062437,4480253,9718457,21055958,45575049,98566055,
%U A362973 213028539,460160083,993533517,2144335391,4626664451,9980028172,21523027285,46408635232,100053270534
%N A362973 The number of cubefull numbers (A036966) not exceeding 10^n.
%C A362973 The number of cubefull numbers not exceeding x is N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)), where c_0 (A362974), c_1 (A362975) and c_2 (A362976) are constants (Bateman and Grosswald, 1958; Finch, 2003).
%C A362973 The digits of a(3k) converge to A362974 as k -> oo. - _Chai Wah Wu_, May 13 2023
%D A362973 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.6.1, pp. 113-115.
%H A362973 Chai Wah Wu, <a href="/A362973/b362973.txt">Table of n, a(n) for n = 0..36</a>
%H A362973 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 A362973 A. Ivić and P. Shiu, <a href="https://doi.org/10.1215/ijm/1256046597">The distribution of powerful integers</a>, Illinois Journal of Mathematics, Vol. 26, No. 4 (1982), pp. 576-590.
%H A362973 Ekkehard Krätzel, <a href="https://doi.org/10.1007/BF01585911">On the distribution of square-full and cube-full numbers</a>, Monatshefte für Mathematik, Vol. 120, No. 2 (1995), pp. 105-119.
%H A362973 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 A362973 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 A362973 There are 2 cubefull numbers not exceeding 10, 1 and 8, therefore a(1) = 2.
%t A362973 a[n_] := Module[{max = 10^n}, CountDistinct@ Flatten@ Table[i^5 * j^4 * k^3, {i, Surd[max, 5]}, {j, Surd[max/i^5, 4]}, {k, CubeRoot[max/(i^5*j^4)]}]]; Array[a, 15, 0]
%o A362973 (Python)
%o A362973 from math import gcd
%o A362973 from sympy import factorint, integer_nthroot
%o A362973 def A362973(n):
%o A362973     m, c = 10**n, 0
%o A362973     for x in range(1,integer_nthroot(m,5)[0]+1):
%o A362973         if all(d<=1 for d in factorint(x).values()):
%o A362973             for y in range(1,integer_nthroot(z:=m//x**5,4)[0]+1):
%o A362973                 if gcd(x,y)==1 and all(d<=1 for d in factorint(y).values()):
%o A362973                     c += integer_nthroot(z//y**4,3)[0]
%o A362973     return c # _Chai Wah Wu_, May 11-13 2023
%Y A362973 Cf. A036966, A362974, A362975, A362976.
%Y A362973 Similar sequences: A070428, A118896.
%K A362973 nonn
%O A362973 0,2
%A A362973 _Amiram Eldar_, May 11 2023
%E A362973 a(23)-a(31) from _Chai Wah Wu_, May 11 2023