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A362973
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The number of cubefull numbers (A036966) not exceeding 10^n.
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4
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1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136, 90619, 198767, 434572, 947753, 2062437, 4480253, 9718457, 21055958, 45575049, 98566055, 213028539, 460160083, 993533517, 2144335391, 4626664451, 9980028172, 21523027285, 46408635232, 100053270534
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OFFSET
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0,2
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COMMENTS
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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).
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.6.1, pp. 113-115.
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LINKS
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EXAMPLE
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There are 2 cubefull numbers not exceeding 10, 1 and 8, therefore a(1) = 2.
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MATHEMATICA
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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]
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PROG
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(Python)
from math import gcd
from sympy import factorint, integer_nthroot
m, c = 10**n, 0
for x in range(1, integer_nthroot(m, 5)[0]+1):
if all(d<=1 for d in factorint(x).values()):
for y in range(1, integer_nthroot(z:=m//x**5, 4)[0]+1):
if gcd(x, y)==1 and all(d<=1 for d in factorint(y).values()):
c += integer_nthroot(z//y**4, 3)[0]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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