A355261 a(n) = largest-nth-power(n, 2) * radical(n) = A000188(n) * A007947(n), where largest-nth-power(n, e) is the largest positive integer b such that b^e divides n.
1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 8, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68
Offset: 1
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
-
Maple
with(NumberTheory): seq(LargestNthPower(n, 2)*Radical(n), n = 1..68);
-
Mathematica
Array[Apply[Times, #[[All, 1]]]*Apply[Times, #1^Floor[#2/2] & @@ Transpose@ #] &@ FactorInteger[#] &, 68] (* Michael De Vlieger, Jul 12 2022 *)
-
Python
from math import prod from sympy import factorint def A355261(n): return prod(p**((e>>1)+1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 13 2022
Formula
Multiplicative with a(p^e) = p^(1+floor(e/2)). - Amiram Eldar, Jul 13 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.447583182004... . - Amiram Eldar, Nov 13 2022