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A355263
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a(n) = largest-nth-power(n, 3) * radical(n) = A053150(n) * A007947(n), where the largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.
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2
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1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 12, 5, 26, 9, 14, 29, 30, 31, 4, 33, 34, 35, 6, 37, 38, 39, 20, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 18, 55, 28, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(1 + floor(e/3)). - Amiram Eldar, Jul 13 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(5)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 2/p^5 + 1/p^6) = 0.3643121583... . - Amiram Eldar, Nov 13 2022
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MAPLE
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with(NumberTheory): seq(LargestNthPower(n, 3)*Radical(n), n=1..69);
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MATHEMATICA
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f[p_, e_] := p^(1 + Floor[e/3]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 13 2022 *)
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PROG
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(Python)
from math import prod
from sympy import factorint
def A355263(n): return prod(p**(e//3+1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 13 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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