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%I #44 Nov 03 2021 06:11:55
%S 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,
%T 10,10,10,11,11,12,12,12,12,12,12,13,13,13,14,14,15,15,15,15,15,15,16,
%U 17,17,17,17,18,18,18,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22
%N 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).
%C 36, 49, 52 and 55 are not in this sequence. - _Don Reble_, Nov 29 2001
%C a(n) >= a(n-1). - Larry Reeves (larryr(AT)acm.org), Jan 06 2005
%C From _Bernard Schott_, Oct 31 2021: (Start)
%C a(n) is a monotonic, though not strictly monotonic, increasing function of n.
%C Complement for 1st comment: a(124) = 35 and a(125) = 37 (see Guy's book). (End)
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B22, pp. 122-123.
%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/1967-a1987-les-factorielles-revisitees">A1987 - Les factorielles revisitées</a>.
%H Richard K. Guy and John L. Selfridge, <a href="http://www.jstor.org/stable/2588996">Factoring factorial n</a>, Amer. Math. Monthly, Vol. 105, No. 8 (1998), pp. 766-767.
%H Bernard Schott, <a href="/A034258/a034258_2.txt">Corresponding products for n!</a>
%F If p is prime, a(p-1) = a(p). - _Bernard Schott_, Oct 24 2021
%e 3! = 6 = 1*2*3 is the only possible factorization, so a(3) = 1.
%e 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.
%Y Cf. A034259, A034260.
%K nonn,nice
%O 1,4
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, May 12 2001
%E Verified by _Don Reble_, Apr 22 2007
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