

A157017


Numbers n such that n! can be written as a product of distinct factors in the range from n+1 to 2n, inclusive.


5



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

1,1


COMMENTS

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


REFERENCES

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), 243257.


LINKS

Table of n, a(n) for n=1..82.
Ray Chandler, Detailed examples for terms in A157017
P. Erdos, Consecutive integers, Eureka, The Archimedeans' Journal, 38 (1975/76), 38.
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)
T. D. Noe, Representations of n!


FORMULA

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


EXAMPLE

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.


PROG

(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, f1, p/f)) && return(1))} \\ M. F. Hasler, Feb 10 2014


CROSSREFS

Cf. A000142, A157625, A157229.
Sequence in context: A184657 A310136 A310137 * A004957 A026352 A198084
Adjacent sequences: A157014 A157015 A157016 * A157018 A157019 A157020


KEYWORD

full,fini,nonn


AUTHOR

Jaume Oliver Lafont, Feb 21 2009


EXTENSIONS

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
Terms 159 to 239 added by Ray Chandler and T. D. Noe, Mar 01 2009
Erroneous term 5 removed by Markus Koenig (markus(AT)stberkoenig.de), Mar 13 2010
Erroneous terms 75 and 88 removed by T. D. Noe, Apr 01 2010
Terms up to 160 doublechecked by M. F. Hasler, Feb 10 2014


STATUS

approved



