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).

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

Offset

1,4

Comments

36, 49, 52 and 55 are not in this sequence. - Don Reble, Nov 29 2001

a(n) >= a(n-1). - 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. 122-123.

Links

Table of n, a(n) for n=1..79.

Diophante, A1987 - Les factorielles revisitées.

Richard K. Guy and John L. Selfridge, Factoring factorial n, Amer. Math. Monthly, Vol. 105, No. 8 (1998), pp. 766-767.

Bernard Schott, Corresponding products for n!

Formula

If p is prime, a(p-1) = 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