

A034258


Write n! as a product of n numbers, n! = k(1)*k(2)*...*k(n) with k(1) <= k(2) <= ..., in all possible ways; a(n) = max value of k(1).


3



1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22
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OFFSET

1,4


COMMENTS

36, 49, 52 and 55 are not in this sequence.  Don Reble, Nov 29 2001
a(n) >= a(n1).  Larry Reeves (larryr(AT)acm.org), Jan 06 2005
From Bernard Schott, Oct 31 2021: (Start)
a(n) is a monotonic, though not strictly monotonic, increasing function of n.
Complement for 1st comment: a(124) = 35 and a(125) = 37 (see Guy's book). (End)


REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B22, pp. 122123.


LINKS

Table of n, a(n) for n=1..79.
Diophante, A1987  Les factorielles revisitÃ©es.
Richard K. Guy and John L. Selfridge, Factoring factorial n, Amer. Math. Monthly, Vol. 105, No. 8 (1998), pp. 766767.
Bernard Schott, Corresponding products for n!


FORMULA

If p is prime, a(p1) = a(p).  Bernard Schott, Oct 24 2021


EXAMPLE

3! = 6 = 1*2*3 is the only possible factorization, so a(3) = 1.
27! = 8^4 * 9^6 * 10^6 * 11^2 * 12 * 13^2 * 14^3 * 17 * 19 * 23, with 4 + 6 + 6 + 2 + 1 + 2 + 3 + 1 + 1 + 1 = 27 factors, which is the required number. Since the first factor is 8, a(27) >= 8. In fact no larger value can be obtained and a(27) = 8.


CROSSREFS

Cf. A034259, A034260.
Sequence in context: A241766 A351646 A025811 * A184349 A290573 A218085
Adjacent sequences: A034255 A034256 A034257 * A034259 A034260 A034261


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Robert G. Wilson v, May 12 2001
Verified by Don Reble, Apr 22 2007


STATUS

approved



