1,1

Erdos remarks that this is a finite sequence. - N. J. A. Sloane (njas(AT)research.att.com), Feb 23 2009

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]

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]

P. Erdos: Consecutive integers, Eureka, The Archimedeans' Journal, 38 (1975/76), 3--8.

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]

T. D. Noe, Representations of n!

Ray Chandler, Detailed examples for terms in A157017

P. Erdos, Consecutive integers (1975)

P. Erdos, Consecutive integers (1975) [Cached copy]

P. Erdos, R. K. Guy and J. L. Selfridge, Another property of 239 and some related questions (1982) [From T. D. Noe (noe(AT)sspectra.com), Mar 01 2009]

3! = 6. [Vaughan, quoted by Erdos]

6! = 8*9*10. [Erdos]

8! = 12*14*15*16. [Vaughan, quoted by Erdos]

11! = 15*16*18*20*21*22. [Vaughan, quoted by Erdos]

14! = 16*21*22*24*25*26*27*28. [Erdos]

15! = 16*18*20*21*22*25*26*27*28. [Vaughan, quoted by Erdos]

18! = 20*21*22*24*26*27*30*32*34*35*36.

18! = 20*21*24*25*26*27*28*32*33*34*36.

18! = 21*22*24*25*26*27*28*30*32*34*36.

21! = 24*25*27*28*32*33*34*35*36*38*39*40*42.

22! = 24*25*26*27*28*30*32*33*34*35*36*38*42*44.

25! = 26*27*30*32*33*34*35*36*38*40*44*45*46*48*49*50.

25! = 27*28*30*32*33*34*35*38*39*40*42*44*45*46*48*50.

28! = 30*32*33*36*38*39*40*42*45*46*48*49*50*51*52*54*55*56.

29! = 30*32*33*34*35*36*39*40*42*44*45*46*48*49*50*52*54*57*58.

29! = 30*32*33*35*36*38*39*40*42*44*45*46*48*49*50*51*52*54*58.

32! = 34*35*36*39*40*42*44*45*46*48*50*52*54*55*56*57*58*60*62*63*64

32! = 35*36*38*39*40*42*44*45*46*48*50*51*52*54*55*56*58*60*62*63*64

35! = 36*40*44*45*48*49*50*51*52*54*55*56*57*58*60*62*63*64*65*66*68*69*70

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

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

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]

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)

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)

See link for further example.

Cf. A000142.

Adjacent sequences: A157014 A157015 A157016 this_sequence A157018 A157019 A157020

Sequence in context: A047444 A160734 A121501 this_sequence A062832 A089085 A033163

full,fini,nonn

Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 21 2009

Gave more precise definition and added the term 18. - R. J. Mathar, Feb 21 2009

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

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.

Ray Chandler and T. D. Noe added terms 159 to 239 T. D. Noe (noe(AT)sspectra.com), Mar 01 2009