A244668 Numerators of (product of divisors of n / sum of divisors of n).

1, 2, 3, 8, 5, 3, 7, 64, 27, 50, 11, 432, 13, 49, 75, 1024, 17, 1944, 19, 4000, 441, 121, 23, 27648, 125, 338, 729, 392, 29, 11250, 31, 32768, 363, 578, 1225, 10077696, 37, 361, 1521, 256000, 41, 64827, 43, 21296, 30375, 529, 47, 63700992, 343, 125000, 289, 70304, 53, 354294, 3025

Offset: 1

Author

Jaroslav Krizek, Jul 04 2014

Keywords

• nonn
• frac

Examples

			a(7) = 7 because the divisors of 7 are 1 and 7, so then (1 * 7) / (1 + 7) = 7 / 8.
a(8) = 64 because the divisors of 8 are 1, 2, 4 and 8, so then (1 * 2 * 4 * 8) / (1 + 2 + 4 + 8) = 64 / 15.
a(9) = 27 because the divisors of 9 are 1, 3 and 9, so then (1 * 3 * 9) / (1 + 3 + 9) = 27 / 13.
		

Crossrefs

Cf. A007955, A000203, A145551, A244669, A244670.

For denominators see A244669.

Programs

Magma

[Numerator((&*[d: d in Divisors(n)]) / (&+[d: d in Divisors(n)])): n in [1..100]]

Mathematica

Table[Numerator[(Times@@Divisors[n])/(Plus@@Divisors[n])], {n, 50}] (* Alonso del Arte, Jul 05 2014 *)

PARI

a007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)) ;
a(n) = numerator(a007955(n)/sigma(n)); \\ Michel Marcus, Jul 05 2014

Formula

Numerators of (A007955(n) / A000203(n)).

a(n) = n if n = 1 or if n is prime.

a(n) < n if and only if n = 6.