A072499 Product of divisors of n which are <= n^(1/2).

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 8, 1, 6, 1, 8, 3, 2, 1, 24, 5, 2, 3, 8, 1, 30, 1, 8, 3, 2, 5, 144, 1, 2, 3, 40, 1, 36, 1, 8, 15, 2, 1, 144, 7, 10, 3, 8, 1, 36, 5, 56, 3, 2, 1, 720, 1, 2, 21, 64, 5, 36, 1, 8, 3, 70, 1, 1152, 1, 2, 15, 8, 7, 36, 1, 320, 27, 2, 1, 1008, 5, 2, 3, 64, 1

Offset: 1

Author

Amarnath Murthy, Jul 20 2002

Keywords

• nonn

Comments

a(1) = 1 and a(24) = 24. For each pair of primes p,q such that p < q < p^2, if n = p^3*q, then a(n) = n. There are others as well; e.g., a(40) = 40. - Don Reble, Aug 02 2002

Row products of the table in A161906. - Reinhard Zumkeller, Mar 08 2013

It appears that the fixed points belong to 3 categories: p^6 (A030516), p^3*q, or p*q*r. - Michel Marcus, May 16 2014

Examples

			a(20) = 8. The divisors of 20 are 1,2,4,5,10 and 20. a(20) = 1*2*4 = 8.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A072500, A072501, A161906.

Cf. A072504, A066839.

Programs

Haskell

a072499 = product . a161906_row  -- Reinhard Zumkeller, Mar 08 2013

Mathematica

a[n_] := Times @@ Select[Divisors[n], #^2 <= n &]; Array[a, 100] (* Amiram Eldar, Jul 31 2022 *)

PARI

a(n) = my(d = divisors(n)); prod(i=1, #d, if (d[i]^2 <= n, d[i], 1)); \\ Michel Marcus, May 16 2014

Python

from math import prod
from itertools import takewhile
from sympy import divisors
def A072499(n): return prod(takewhile(lambda x:x**2<=n,divisors(n))) # Chai Wah Wu, Dec 19 2023

Extensions

More terms from Sascha Kurz, Feb 02 2003