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A188902 Numerator of the base n logarithm of the product of the divisors of n. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A256262 A332929

Adjacent sequences:  A188899 A188900 A188901 * A188903 A188904 A188905

KEYWORD

nonn,easy,frac

AUTHOR

Alonso del Arte, Apr 19 2011

STATUS

approved

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Last modified May 29 16:43 EDT 2020. Contains 334704 sequences. (Running on oeis4.)