A157017 Numbers n such that n! can be written as a product of distinct factors in the range from n+1 to 2n, inclusive.

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

Offset: 1

Author

Jaume Oliver Lafont, Feb 21 2009

Keywords

• full
• fini
• nonn

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

Examples

			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.
		

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), 243-257.

Links

• Ray Chandler, Detailed examples for terms in A157017
• P. Erdos, Consecutive integers, Eureka, The Archimedeans' Journal, 38 (1975/76), 3-8.
• 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!

Crossrefs

Cf. A000142, A157625, A157229.

Programs

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

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

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)stber-koenig.de), Mar 13 2010

Erroneous terms 75 and 88 removed by T. D. Noe, Apr 01 2010

Terms up to 160 double-checked by M. F. Hasler, Feb 10 2014