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

1,2

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

Links

Table of n, a(n) for n=1..47.

Index entries for two-way infinite sequences.

Formula

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

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

Example

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.

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 *)

Prog

(Python)
from sympy import divisor_count
def A056925(n): return n**(divisor_count(n)//2) # Chai Wah Wu, Jun 25 2022
(PARI) a(n) = n^(numdiv(n)\2); \\ Michel Marcus, Nov 11 2023

Crossrefs

Cf. A007955, A056924.

Sequence in context: A061537 A306329 A274029 * A060067 A043310 A171578

Adjacent sequences: A056922 A056923 A056924 * A056926 A056927 A056928

Keyword

nonn

Author

Henry Bottomley, Jul 12 2000

Status

approved