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A286981
Binomial coefficients binomial(n,k) = UV such that n>=2k and U > V, where gpf(U) <= k, gpf(V) > k (gpf(n)= is the greatest prime factor of n).
1
56, 84, 120, 126, 252, 792, 816, 88560, 98280, 116280, 203490, 695520, 2035800, 177100560, 573166440, 818809200, 2310789600, 8597496600, 1889912732400
OFFSET
1,1
COMMENTS
The corresponding pairs (n,k) are: (8,3), (9,3), (10,3), (9,4), (10,5), (12,5), (18,3), (82,3), (28,5), (21,7), (21,8), (162,3), (30,7), (54,7), (33,13), (33,14), (36,13), (36,17), (56,13).
Ecklund et al. proved that the sequence is finite, and that these are the only terms, except for the cases k = 3, 5 and 7, but they strongly conjectured that the list is complete. They also give the near miss binomial(514,3)=22500864=UV, with U=2^9*3^2=4608, V=19*257=4883, and V/U < 1.06.
No other terms below 10^20.
Supersequence of A286980.
LINKS
Earl F. Ecklund, Jr., Roger B. Eggleton, Paul Erdős and John L. Selfridge, On the prime factorization of binomial coefficients, Journal of the Australian Mathematical Society (Series A), Vol. 26, No. 3 (1978), pp. 257-269.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, B31, p. 130.
EXAMPLE
84 = Binomial(9,3) = 12*7, gpf(12) = 3 <= 3 and gpf(7) = 7 > 3, and 12 > 7, thus 84 is in the sequence.
CROSSREFS
Sequence in context: A043938 A328165 A214250 * A254369 A234927 A104394
KEYWORD
nonn,fini
AUTHOR
Amiram Eldar, May 17 2017
STATUS
approved