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Search: id:A100933
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| A100933 |
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Numbers having more than one representation as the product of consecutive integers > 1. |
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+0
3
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| 120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, 368406749739154248105984000000
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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
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REFERENCES
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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.
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LINKS
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Robert Munafo's page on A100933
Robert Munafo, Page on A100933.
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EXAMPLE
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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.
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MATHEMATICA
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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]
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Nov 22 2004
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EXTENSIONS
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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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