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A168363
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Squares and cubes of primes.
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12
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4, 8, 9, 25, 27, 49, 121, 125, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 17161, 18769, 19321, 22201
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OFFSET
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1,1
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COMMENTS
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Primitive elements for powerful numbers; every powerful is product of these numbers. The representation is not necessarily unique.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = P(2) + P(3) = 0.6270100593..., where P is the prime zeta function. - Amiram Eldar, Dec 21 2020
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MATHEMATICA
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With[{nn=50}, Take[Union[Flatten[Table[{n^2, n^3}, {n, Prime[Range[ nn]]}]]], nn]] (* Harvey P. Dale, Feb 26 2015 *)
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PROG
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(PARI) for(n=1, 40000, fm=factor(n); if(matsize(fm)[1]==1&(fm[1, 2]==2||fm[1, 2]==3), print1(n", ")))
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot
def f(x): return n+x-primepi(isqrt(x))-primepi(integer_nthroot(x, 3)[0])
m, k = n, f(n)
while m != k:
m, k = k, f(k)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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