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, 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
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
-
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 *) -
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
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)