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A286981
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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).
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1
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56, 84, 120, 126, 252, 792, 816, 88560, 98280, 116280, 203490, 695520, 2035800, 177100560, 573166440, 818809200, 2310789600, 8597496600, 1889912732400
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, B31, p. 130.
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EXAMPLE
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84 = Binomial(9,3) = 12*7, gpf(12) = 3 <= 3 and gpf(7) = 7 > 3, and 12 > 7, thus 84 is in the sequence.
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CROSSREFS
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KEYWORD
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nonn,fini
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AUTHOR
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STATUS
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approved
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