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A157017
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Numbers n such that n! can be written as a product of distinct factors in the range from n+1 to 2n, inclusive.
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5
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3, 6, 8, 11, 14, 15, 18, 21, 22, 25, 28, 29, 32, 35, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 61, 63, 64, 67, 68, 69, 73, 74, 76, 77, 78, 86, 89, 90, 94, 95, 98, 99, 103, 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
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OFFSET
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1,1
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COMMENTS
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Erdos remarks that this is a finite sequence. - N. J. A. Sloane, 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, 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
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
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REFERENCES
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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.
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LINKS
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FORMULA
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EXAMPLE
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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 = cp(18) / (25*28*33).
18! = 20*21*24*25*26*27*28*32*33*34*36 = cp(18) / (22*30*35).
18! = 21*22*24*25*26*27*28*30*32*34*36 = cp(18) / (20*33*35).
21! = 24*25*27*28*32*33*34*35*36*38*39*40*42 = cp(21) / (22*26*30).
22! = 24*25*26*27*28*30*32*33*34*35*36*38*42*44 = cp(22) / (39*40).
25! = 26*27*30*32*33*34*35*36*38*40*44*45*46*48*49*50 = cp(25) / (28*39*42).
25! = 27*28*30*32*33*34*35*38*39*40*42*44*45*46*48*50 = cp(25) / (26*36*49).
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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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full,fini,nonn
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AUTHOR
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EXTENSIONS
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Erroneous term 5 removed by Markus Koenig (markus(AT)stber-koenig.de), Mar 13 2010
Erroneous terms 75 and 88 removed by T. D. Noe, Apr 01 2010
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STATUS
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approved
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