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%I #26 Apr 25 2019 03:33:21
%S 3,6,8,11,14,15,18,21,22,25,28,29,32,35,39,40,43,44,47,48,51,52,55,56,
%T 59,60,61,63,64,67,68,69,73,74,76,77,78,86,89,90,94,95,98,99,103,104,
%U 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 Numbers n such that n! can be written as a product of distinct factors in the range from n+1 to 2n, inclusive.
%C Erdos remarks that this is a finite sequence. - _N. J. A. Sloane_, Feb 23 2009
%C 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, including the ends (A157625). 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. - _T. D. Noe_, Feb 25 2009
%C 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. - _T. D. Noe_, Mar 01 2009
%D 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.
%H Ray Chandler, <a href="/A157017/a157017b.txt">Detailed examples for terms in A157017</a>
%H P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1975-07.pdf">Consecutive integers</a>, Eureka, The Archimedeans' Journal, 38 (1975/76), 3-8.
%H P. Erdos, <a href="/A157017/a157017.pdf">Consecutive integers</a> (1975) [Cached copy]
%H 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)
%H T. D. Noe, <a href="/A157017/a157017.txt">Representations of n!</a>
%F A number n is in the sequence iff A000984(n)*A034386(n)/A034386(2n) is the product of distinct composite numbers in {n+1,...,2n}. - _M. F. Hasler_, Feb 10 2014
%e 3! = 6. [Vaughan, quoted by Erdos]
%e 6! = 8*9*10. [Erdos]
%e 8! = 12*14*15*16. [Vaughan, quoted by Erdos]
%e 11! = 15*16*18*20*21*22. [Vaughan, quoted by Erdos]
%e 14! = 16*21*22*24*25*26*27*28. [Erdos]
%e 15! = 16*18*20*21*22*25*26*27*28. [Vaughan, quoted by Erdos]
%e 18! = 20*21*22*24*26*27*30*32*34*35*36 = cp(18) / (25*28*33).
%e 18! = 20*21*24*25*26*27*28*32*33*34*36 = cp(18) / (22*30*35).
%e 18! = 21*22*24*25*26*27*28*30*32*34*36 = cp(18) / (20*33*35).
%e 21! = 24*25*27*28*32*33*34*35*36*38*39*40*42 = cp(21) / (22*26*30).
%e 22! = 24*25*26*27*28*30*32*33*34*35*36*38*42*44 = cp(22) / (39*40).
%e 25! = 26*27*30*32*33*34*35*36*38*40*44*45*46*48*49*50 = cp(25) / (28*39*42).
%e 25! = 27*28*30*32*33*34*35*38*39*40*42*44*45*46*48*50 = cp(25) / (26*36*49).
%e 28! = 30*32*33*36*38*39*40*42*45*46*48*49*50*51*52*54*55*56.
%e 29! = 30*32*33*34*35*36*39*40*42*44*45*46*48*49*50*52*54*57*58.
%e 29! = 30*32*33*35*36*38*39*40*42*44*45*46*48*49*50*51*52*54*58.
%e 32! = 34*35*36*39*40*42*44*45*46*48*50*52*54*55*56*57*58*60*62*63*64
%e 32! = 35*36*38*39*40*42*44*45*46*48*50*51*52*54*55*56*58*60*62*63*64
%e 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 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 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 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 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 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 See link for further example.
%o (PARI) is_A237594(n,m=2*n,p=binomial(2*n,n)/prod(k=primepi(n)+1,primepi(n*2),prime(k)))={forstep(f=m,n+1,-1, p%f==0 && (p==f || is_A237594(n,f-1,p/f)) && return(1))} \\ _M. F. Hasler_, Feb 10 2014
%Y Cf. A000142, A157625, A157229.
%K full,fini,nonn
%O 1,1
%A _Jaume Oliver Lafont_, Feb 21 2009
%E More precise definition and term 18 from _R. J. Mathar_, Feb 21 2009 Terms 21 through 73 added by _Ray Chandler_ and _T. D. Noe_, further terms up to 158 by _T. D. Noe_, Feb 24 2009
%E Terms 159 to 239 added by _Ray Chandler_ and _T. D. Noe_, Mar 01 2009
%E Erroneous term 5 removed by Markus Koenig (markus(AT)stber-koenig.de), Mar 13 2010
%E Erroneous terms 75 and 88 removed by _T. D. Noe_, Apr 01 2010
%E Terms up to 160 double-checked by _M. F. Hasler_, Feb 10 2014
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