A072501 Ratio of the product of divisors of n which are > n^(1/2) to product of divisors of n which are < n^(1/2).

1, 2, 3, 4, 5, 9, 7, 16, 9, 25, 11, 48, 13, 49, 25, 64, 17, 162, 19, 125, 49, 121, 23, 576, 25, 169, 81, 343, 29, 900, 31, 512, 121, 289, 49, 2916, 37, 361, 169, 1600, 41, 2401, 43, 1331, 405, 529, 47, 12288, 49, 1250, 289, 2197, 53, 6561, 121, 3136, 361, 841, 59

Offset: 1

Author

Amarnath Murthy, Jul 20 2002

Keywords

• nonn

Comments

It can easily be proved that the ratio is always an integer. a(n) = n if n is a prime or the square of a prime.

If 1/3 were chosen as the exponent instead of 1/2, then the sequence would begin: 1, 2, 3, 8, 5, 36, 7, 32, 27, .... If the exponent is decreased along 1/4, 1/5, ..., then the resulting sequence tends towards A007955. - Michel Marcus, Sep 17 2013

Examples

			a(20) = 25. The divisors of 20 are 1,2,4,5,10 and 20. a(20) = 10*20/2*4 = 25.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Ratio of corresponding terms of A072500 and A072499.

Programs

Mathematica

Table[Times @@ ((d = Divisors[n])^Sign[d - Sqrt[n]]), {n, 1, 59}] (* Ivan Neretin, May 01 2016 *)

PARI

a(n) = {d = divisors(n); pa = 1; pb = 1; fordiv(n, d, if (d^2 < n, pa *= d); if (d^2 > n, pb *= d);); pb/pa;} \\ Michel Marcus, Sep 17 2013

Extensions

More terms from Sascha Kurz, Feb 02 2003