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A118896
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Number of powerful numbers <= 10^n.
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5
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1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, 6840384, 21663503, 68575557, 217004842, 686552743, 2171766332, 6869227848, 21725636644, 68709456167, 217293374285, 687174291753, 2173105517385, 6872112993377, 21731852479862, 68722847672629, 217322225558934, 687236449779456, 2173239433013146
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OFFSET
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0,2
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COMMENTS
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These numbers agree with the asymptotic formula c*sqrt(x), with c=2.1732...(A090699). - T. D. Noe, May 09 2006
Bateman & Grosswald proved that the number of powerful numbers up to x is zeta(3/2)/zeta(3) * x^1/2 + zeta(2/3)/zeta(2) * x^1/3 + o(x^1/6). This approximates the series very closely: up to a(24), all absolute errors are less than 75. - Charles R Greathouse IV, Sep 23 2008
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LINKS
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FORMULA
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Pi(x) = Sum_{i=1..x^(1/3)} floor(sqrt(x/i^3)) only if i is squarefree. - Robert G. Wilson v, Aug 12 2014
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MAPLE
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f:= m -> nops({seq(seq(a^2*b^3, b=1..floor((m/a^2)^(1/3))), a=1..floor(sqrt(m)))}):
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MATHEMATICA
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f[n_] := Block[{max = 10^n}, Length@ Union@ Flatten@ Table[ a^2*b^3, {b, max^(1/3)}, {a, Sqrt[ max/b^3]}]]; Array[f, 13, 0] (* Robert G. Wilson v, Aug 11 2014 *)
powerfulNumberPi[n_] := Sum[ If[ SquareFreeQ@ i, Floor[ Sqrt[ n/i^3]], 0], {i, n^(1/3)}]; Array[ powerfulNumberPi[10^#] &, 27, 0] (* Robert G. Wilson v, Aug 12 2014 *)
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PROG
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(PARI) a(n)=n=10^n; sum(k=1, floor((n+.5)^(1/3)), if(issquarefree(k), sqrtint(n\k^3))) \\ Charles R Greathouse IV, Sep 23 2008
(Python)
from math import isqrt
from sympy import integer_nthroot, factorint
m = 10**n
return sum(isqrt(m//x**3) for x in range(1, integer_nthroot(m, 3)[0]+1) if max(factorint(x).values(), default=0)<=1) # Chai Wah Wu, May 13 2023
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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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