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 ErdÅ‘s 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

KEYWORD

nonn,fini

AUTHOR

Amiram Eldar, May 17 2017

STATUS

approved