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A355264
a(n) = n * largest-nth-power(n, 2) = n * A000188(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.
1
1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 24, 13, 14, 15, 64, 17, 54, 19, 40, 21, 22, 23, 48, 125, 26, 81, 56, 29, 30, 31, 128, 33, 34, 35, 216, 37, 38, 39, 80, 41, 42, 43, 88, 135, 46, 47, 192, 343, 250, 51, 104, 53, 162, 55, 112, 57, 58, 59, 120, 61, 62, 189, 512
OFFSET
1,2
LINKS
FORMULA
Multiplicative with a(p^e) = p^(e+floor(e/2)). - Amiram Eldar, Jul 13 2022
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(2*s-3)/ zeta(2*s-2).
Sum_{k=1..n} a(k) ~ (3*n^2/(4*Pi^2)) * (2*log(n) + 6*gamma - 4*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620). (End)
MAPLE
with(NumberTheory): seq(n*LargestNthPower(n, 2), n = 1..64);
MATHEMATICA
Table[n*Times @@ (#1^Floor[#2/2] & @@@ FactorInteger[n]), {n, 64}] (* Michael De Vlieger, Jul 12 2022 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(f[i, 2] + f[i, 2]\2)); } \\ Amiram Eldar, Sep 21 2023
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Peter Luschny, Jul 12 2022
STATUS
approved