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).

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

Table of n, a(n) for n=1..19.

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

Cf. A007318, A286980.

Sequence in context: A043938 A328165 A214250 * A254369 A234927 A104394

Adjacent sequences: A286978 A286979 A286980 * A286982 A286983 A286984

Keyword

nonn,fini

Author

Amiram Eldar, May 17 2017

Status

approved