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A075760 Nontrivial binomial coefficients which are perfect powers (A001597). 0
36, 1225, 19600, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Triangular-square numbers (A001110) are a subset, except for 0 and 1.
"For C(n,k) k>=4 and any l>=2 no solutions exist and this is what Erdos proved by an ingenious argument. ... C(50, 3) = 140^2 is the only solution for k = 3, l=2." page 13 of Aigner and Ziegler.
REFERENCES
Martin Aigner and Gunter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin, 2000, Chapter 3, "Binomial coefficients are (almost) never powers," pages 13-16.
LINKS
MATHEMATICA
f[n_] := Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]]; a = {}; Do[ If[ f[n(n - 1)/2] > 1, a = Append[a, Binomial[n, 2]]]; If[ f[n(n - 1)*(n - 2)/6] > 1, a = Append[a, Binomial[n, 3]]], {n, 5, 1500000}]
CROSSREFS
Cf. A001110.
Sequence in context: A215858 A103278 A004294 * A113938 A001110 A331234
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Oct 08 2002
STATUS
approved

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Last modified February 24 17:27 EST 2024. Contains 370307 sequences. (Running on oeis4.)