%I A157017
%S A157017 3,5,6,8,11,14,15,18,21,22,25,28,29,32,35,39,40,43,44,47,48,51,52,55,56,
%T A157017 59,60,61,63,64,67,68,69,73,74,75,76,77,78,86,88,89,90,94,95,98,99,103,
%U A157017 104,107,116,117,122,123,124,125,126,127,131,145,146,149,158,159,179,183,
               187,188,189,191,194,203,207,215,218,219,221,222,223,224,229,230,233,
               238,239
%N A157017 Numbers n such that n! can be written as a product of distinct factors 
               in the range from n+1 to 2n, inclusive.
%C A157017 Erdos remarks that this is a finite sequence. - N. J. A. Sloane (njas(AT)research.att.com), 
               Feb 23 2009
%C A157017 Here is another way of displaying a representation of n!: Let cp(n) be 
               the product of the composite numbers from n+1 to 2n. For example, 
               40! = cp(40) / (46*70*77). Because the number of factors in the denominator 
               is small relative to n, this simpler form gives us a fast method 
               of finding representations of n!: find distinct factors of cp(n)/
               n! among the numbers n+1 to 2n. See A157229 for the number of representations 
               of n! for the n in this sequence. [From T. D. Noe (noe(AT)sspectra.com), 
               Feb 25 2009]
%C A157017 Erdos et al. found this sequence and showed that 239 is the last term. 
               Note that 239! has 94766 representations! Sequence A157229, which 
               is also in the Erdos et al. paper, gives the number of representations 
               for each n. Ray Chandler and I created an algorithm that verifies 
               the numbers in both sequences. [From T. D. Noe (noe(AT)sspectra.com), 
               Mar 01 2009]
%D A157017 P. Erdos: Consecutive integers, Eureka, The Archimedeans' Journal, 38 
               (1975/76), 3--8.
%D A157017 P. Erdos, R. K. Guy and J. L. Selfridge, Another property of 239 and 
               some related questions, Proceedings of the Eleventh Manitoba Conference 
               on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. 
               Numer. 34 (1982), 243-257 [From T. D. Noe (noe(AT)sspectra.com), 
               Mar 01 2009]
%H A157017 T. D. Noe, <a HREF="a157017.txt">Representations of n!</a>
%H A157017 Ray Chandler, <a HREF="a157017b.txt">Detailed examples for terms in A157017</
               a>
%H A157017 P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1975-07.pdf">Consecutive 
               integers</a> (1975)
%H A157017 P. Erdos, <a href="a157017.pdf">Consecutive integers</a> (1975) [Cached 
               copy]
%H A157017 P. Erdos, R. K. Guy and J. L. Selfridge, <a href="http://www.renyi.hu/
               ~p_erdos/1982-01.pdf">Another property of 239 and some related questions</
               a> (1982) [From T. D. Noe (noe(AT)sspectra.com), Mar 01 2009]
%e A157017 3! = 6. [Vaughan, quoted by Erdos]
%e A157017 6! = 8*9*10. [Erdos]
%e A157017 8! = 12*14*15*16. [Vaughan, quoted by Erdos]
%e A157017 11! = 15*16*18*20*21*22. [Vaughan, quoted by Erdos]
%e A157017 14! = 16*21*22*24*25*26*27*28. [Erdos]
%e A157017 15! = 16*18*20*21*22*25*26*27*28. [Vaughan, quoted by Erdos]
%e A157017 18! = 20*21*22*24*26*27*30*32*34*35*36.
%e A157017 18! = 20*21*24*25*26*27*28*32*33*34*36.
%e A157017 18! = 21*22*24*25*26*27*28*30*32*34*36.
%e A157017 21! = 24*25*27*28*32*33*34*35*36*38*39*40*42.
%e A157017 22! = 24*25*26*27*28*30*32*33*34*35*36*38*42*44.
%e A157017 25! = 26*27*30*32*33*34*35*36*38*40*44*45*46*48*49*50.
%e A157017 25! = 27*28*30*32*33*34*35*38*39*40*42*44*45*46*48*50.
%e A157017 28! = 30*32*33*36*38*39*40*42*45*46*48*49*50*51*52*54*55*56.
%e A157017 29! = 30*32*33*34*35*36*39*40*42*44*45*46*48*49*50*52*54*57*58.
%e A157017 29! = 30*32*33*35*36*38*39*40*42*44*45*46*48*49*50*51*52*54*58.
%e A157017 32! = 34*35*36*39*40*42*44*45*46*48*50*52*54*55*56*57*58*60*62*63*64
%e A157017 32! = 35*36*38*39*40*42*44*45*46*48*50*51*52*54*55*56*58*60*62*63*64
%e A157017 35! = 36*40*44*45*48*49*50*51*52*54*55*56*57*58*60*62*63*64*65*66*68*69*70
%e A157017 39! = 40*42*45*48*51*52*54*55*56*57*58*60*62*63*64*65*66*68*69*70*72*74*75*76*77*78
%e A157017 39! = 42*44*45*48*50*51*52*54*56*57*58*60*62*63*64*65*66*68*69*70*72*74*75*76*77*78
%e A157017 40! = 42*44*45*48*49*50*51*52*54*55*56*57*58*60*62*63*64*65*66*68*69*72*74*75*76*78*80. 
               [Vaughan, quoted by Erdos]
%e A157017 43! = 44*48*49*50*52*54*57*58*60*62*63*64*65*66*68*69*70*72*74*75*76*77*78*80*81*82*84*85*86 
               (and 2 other ways)
%e A157017 44! = 45*46*48*49*50*51*52*54*55*56*57*60*62*64*65*66*70*72*74*76*77*78*80*81*82*84*85*86*87*88 
               (and 16 other ways)
%e A157017 See link for further example.
%Y A157017 Cf. A000142.
%Y A157017 Sequence in context: A047444 A160734 A121501 this_sequence A062832 A089085 
               A033163
%Y A157017 Adjacent sequences: A157014 A157015 A157016 this_sequence A157018 A157019 
               A157020
%K A157017 full,fini,nonn
%O A157017 1,1
%A A157017 Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 21 2009
%E A157017 Gave more precise definition and added the term 18. - R. J. Mathar, Feb 
               21 2009
%E A157017 40 is also a member [Vaughan, quoted by Erdos] (but may not be the next 
               term) - Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 21 
               2009
%E A157017 Updates Feb 24 2009: Terms 21 through 73 added by Ray Chandler (rayjchandler(AT)sbcglobal.net) 
               and T. D. Noe (noe(AT)sspectra.com), and further terms up to 158 
               by T. D. Noe.
%E A157017 Ray Chandler and T. D. Noe added terms 159 to 239 T. D. Noe (noe(AT)sspectra.com), 
               Mar 01 2009