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
Terence Tao, Table of n, a(n) for n = 1..10000
Thomas Bloom, Problem 391, Erdős Problems.
Diophante, A1987 - Les factorielles revisitées (c. 2011).
Erdős problems database contributors, Erdős problem database, see no. 391.
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!
Terence Tao, Decomposing a factorial into large factors, arXiv:2503.20170 [math.NT], 2025.
Terence Tao, Erdos-Guy-Selfridge project - a collaborative project to compute upper and lower bounds on this sequence
Terence Tao, The first 10000 elements of the sequence (from the above project)
FORMULA
If p is prime, a(p-1) = a(p). - Bernard Schott, Oct 24 2021
Tao proves that a(n) ~ n/e. Apparently this was in the unpublished work of Erdős, Selfridge, and Straus but this proof has been lost. See Tao (2025) for more precise results. - Charles R Greathouse IV, May 13 2025
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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, May 12 2001
Verified by Don Reble, Apr 22 2007
STATUS
approved