for
an odd prime (Hardy and Wright 1979, p. 98). Actually,
this relationship holds for some composite values as well. Value for which it holds
are ,
3, 4, 5, 6, 7, 10, 11, 13, 17, 19, 23, 29, ... (OEIS A158008).
This can be generalized as follows. Let be an odd primedivisor
of
and
the highest power which divides , then
(3)
and, in particular,
(4)
Now, if
is even and is the highest power of 2 that
divides ,
then
Bauer. Nouvelles annales2, 256-264, 1902.Hardy, G. H. and Wright, E. M. J. London Math. Soc.9, 38-41 and
240, 1934.Hardy, G. H. and Wright, E. M. "Bauer's Identical
Congruence." §8.5 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 98-100, 1979.Sloane, N. J. A. Sequence A158008 in "The On-Line Encyclopedia of Integer
Sequences."