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

2,3

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

Formula

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A080131 A319956 A082882 * A324081 A352522 A256262

Adjacent sequences: A188899 A188900 A188901 * A188903 A188904 A188905

Keyword

nonn,easy,frac

Author

Alonso del Arte, Apr 19 2011

Status

approved