A355261 - OEIS

%I #25 Nov 13 2022 08:40:04

%S 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,

%T 9,28,29,30,31,8,33,34,35,36,37,38,39,20,41,42,43,44,45,46,47,24,49,

%U 50,51,52,53,18,55,28,57,58,59,60,61,62,63,16,65,66,67,68

%N 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.

%H Michael De Vlieger, <a href="/A355261/b355261.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^(1+floor(e/2)). - _Amiram Eldar_, Jul 13 2022

%F 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

%p with(NumberTheory): seq(LargestNthPower(n, 2)*Radical(n), n = 1..68);

%t Array[Apply[Times, #[[All, 1]]]*Apply[Times, #1^Floor[#2/2] & @@ Transpose@ #] &@ FactorInteger[#] &, 68] (* _Michael De Vlieger_, Jul 12 2022 *)

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A355261(n): return prod(p**((e>>1)+1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Jul 13 2022

%Y Cf. A000188, A002117, A007947, A064549, A355263.

%K nonn,mult

%O 1,2

%A _Peter Luschny_, Jul 12 2022