A056925 Largest integer power of n which divides product of divisors of n.

1, 2, 3, 4, 5, 36, 7, 64, 9, 100, 11, 1728, 13, 196, 225, 256, 17, 5832, 19, 8000, 441, 484, 23, 331776, 25, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, 1679616, 37, 1444, 1521, 2560000, 41, 3111696, 43, 85184, 91125, 2116, 47

Offset: 1

Author

Henry Bottomley, Jul 12 2000

Keywords

• nonn

Comments

Product of the distinct parts in the divisor pairs (d,n/d) of n, where d < n/d. For example, the divisors of n = 4 are {1,2,4} with divisor pairs (1,4) and (2,2), but only the pair (1,4) has distinct parts, so a(4) = 1*4 = 4. - Wesley Ivan Hurt, Nov 10 2023

Examples

			a(16)=256 since the factors of 16 are 1,2,4,8,16, their product is 1024 and the largest power of 16 which divides 1024 is 256.
		

Links

• Index entries for two-way infinite sequences.

Crossrefs

Cf. A007955, A056924.

Programs

Mathematica

lip[n_]:=Module[{pr=Times@@Divisors[n],pwr},pwr= Floor[ Log[n,pr]]; n^Last[Select[Range[pwr],Divisible[pr,n^#]&]]]; Join[{1},lip/@ Range[2,50]] (* Harvey P. Dale, Apr 02 2011 *)
a[n_] := n^Floor[DivisorSigma[0, n]/2]; Array[a, 50] (* Amiram Eldar, Jun 26 2022 *)

PARI

a(n) = n^(numdiv(n)\2); \\ Michel Marcus, Nov 11 2023

Python

from sympy import divisor_count
def A056925(n): return n**(divisor_count(n)//2) # Chai Wah Wu, Jun 25 2022

Formula

a(n) = n^A056924(n).

If n is a square a(n) = A007955(n)/sqrt(n), otherwise a(n) = A007955(n).