A188902 Numerator of the base n logarithm of the product of the divisors of n.

1, 1, 3, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 9, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 3, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2

Offset: 2

Author

Alonso del Arte, Apr 19 2011

Keywords

• nonn
• easy
• frac

Comments

Obviously the product of divisors of n (see A007955) is a multiple of n. But often it is also a perfect power of n, a number of the form n^m with m an integer. But if n is a perfect square (A000290), then the logarithm is a rational number but not an integer.

a(1) is of course indeterminate since it can be any value desired, whether real, imaginary or complex.

The denominator is A010052(n) + 1.

Links

• Antti Karttunen, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A000005, A007955, A007956, A038548, A056924, A010052.

Programs

Mathematica

Numerator[Table[FullSimplify[Log[n, Times@@Divisors[n]]], {n, 2, 75}]]

PARI

A188902(n) = numerator(numdiv(n)/2); \\ Antti Karttunen, May 27 2017

Python

from sympy import divisor_count, Integer
def a(n): return (divisor_count(n) / 2).numerator
print([a(n) for n in range(2, 51)])  # Indranil Ghosh, May 27 2017

Formula

a(n) = numerator(A000005(n)/2).

a(n) = (A038548(n) + A056924(n)) / 2 for n > 1.