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A100933 Numbers having more than one representation as the product of consecutive integers > 1. +0
3
120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, 368406749739154248105984000000 (list; graph; listen)
OFFSET

1,1

COMMENT

No others < 10^14. Each of these numbers has two representations. Is two the maximum possible? The sequence is infinite; for any n, the number n*(n+1)*...*(n^2+n-1) is in this sequence. The next number of this form is 20274183401472000, which is obtained when n=4.

I have performed an exhaustive search up to 5.07 x 10^22 and can confirm the next two terms are 20274183401472000 and 25852016738884976640000. No other terms below 5.07 * 10^22 - Robert Munafo (mrob27(AT)gmail.com), Aug 13 2007

Using an improved algorithm I have performed an exhaustive search up to 2.15 * 10^33 and can confirm the terms shown above are all that exist up to that point. For any member of A045619 we can construct a member of this sequence by equating n(n+1)(n+2)...(x-1) to (n+2)(n+3)...(x-1)x. Also, as demonstrated in my examples, 5040 is related to 720 as 259459200 is to 210. So we also know that 36055954861352887137197787308347629783163600896000000000 and 6244042313569035223343873483125151604764341428027427022254596874567680000000000000 are terms. - Robert Munafo (mrob27(AT)gmail.com), Aug 17 2007

REFERENCES

H. L. Abbott, P. Erdos, and D Hanson, On the number of times an integer occurs as a binomial coefficient, Amer. Math. Monthly, Vol. 81, No. 3 (Mar., 1974), 256-261.

LINKS

Robert Munafo's page on A100933

Robert Munafo, Page on A100933.

EXAMPLE

120 is here because 120 = 2*3*4*5 = 4*5*6.

a(2)=210 because we can write 210=5*6*7 or 14*15. The term a(8) = 259459200 = 5*6*7*8*9*10*11*12*13 = 8*9*10*11*12*13*14*15 is related to 210 by adding the intervening integers (8 through 13) to both products.

MATHEMATICA

nn=10^10; t3={}; Do[m=0; p=n; While[m++; p=p(n+m); p<=nn, t3={t3, p}], {n, 2, Sqrt[nn]}]; t3=Sort[Flatten[t3]]; lst={}; Do[If[t3[[i]]==t3[[i+1]], AppendTo[lst, t3[[i]]]], {i, Length[t3]-1}]; Union[lst]

CROSSREFS

Cf. A003015 (numbers occurring 5 or more times in Pascal's triangle).

Cf. A002378, A045619.

Sequence in context: A056994 A114823 A069790 this_sequence A069674 A003015 A098565

Adjacent sequences: A100930 A100931 A100932 this_sequence A100934 A100935 A100936

KEYWORD

nonn

AUTHOR

T. D. Noe (noe(AT)sspectra.com), Nov 22 2004

EXTENSIONS

More terms from Robert Munafo (mrob27(AT)gmail.com), Aug 13 2007

One more term from Robert Munafo (mrob27(AT)gmail.com), Aug 17 2007

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Last modified July 25 07:41 EDT 2008. Contains 142293 sequences.


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