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%I #16 Oct 05 2023 15:46:15
%S 19,34,41,89,104,359,398,495,527,1845,2309,2729,3539,4619,8644,12923,
%T 14135,15774,36098,36569
%N Values of m such that binomial(m, a) + binomial(m, b) divides binomial(m, a + b) for some distinct nonnegative integers a and b with a + b <= m.
%C This sequence was first suggested by _Stefan Steinerberger_, who conjectures that there are infinitely many terms.
%C For every fixed pair of integers a,b, there exists only a finite number of suitable values of m. - _Max Alekseyev_
%C For all known entries (up to a(20)), there is exactly one pair (a,b) which satisfies the required conditions. In every case b-a is either 1 or 2. See sequence A140602 for values with b-a = 1 and A140603 for b-a = 2. Related open questions: (1) Must the pair (a,b) for a given a(n) be unique? (2) Does every solution have b-a <= 2?
%e C(19,3) + C(19,5) divides C(19,8)
%e C(34,6) + C(34,7) divides C(34,13)
%e C(41,5) + C(41,7) divides C(41,12)
%e C(89,7) + C(89,8) divides C(89,15)
%e C(104,3) + C(104,4) divides C(104,7)
%e C(359,5) + C(359,6) divides C(359,11)
%e C(398,20) + C(398,21) divides C(398,41)
%e C(495,12) + C(495,14) divides C(495,26)
%e C(527,7) + C(527,9) divides C(527,16)
%e C(1845,15) + C(1845,17) divides C(1845,32)
%e C(2309,5) + C(2309,6) divides C(2309,11)
%e C(2729,19) + C(2729,20) divides C(2729,39)
%e C(3539,35) + C(3539,36) divides C(3539,71)
%e C(4619,11) + C(4619,12) divides C(4619,23)
%e C(8644,18) + C(8644,19) divides C(8644,37)
%e C(12923,34) + C(12923,36) divides C(12923,70)
%e C(14135,30) + C(14135,31) divides C(14135,61)
%e C(15774,24) + C(15774,26) divides C(15774,50)
%e C(36098,34) + C(36098,36) divides C(36098,70), and C(36569,47) + C(36569,48) divides C(36569,95). - _Robin Visser_, Sep 30 2023
%o (Sage)
%o for m in range(2, 100000):
%o for a in range(0, m//2):
%o for b in range(a+1, m-a+1):
%o if (binomial(m,a+b)%(binomial(m, a)+binomial(m, b)) == 0):
%o print(m) # _Robin Visser_, Sep 30 2023
%Y Cf. A140602, A140603.
%K hard,more,nonn
%O 1,1
%A _Andrew V. Sutherland_, May 18 2008
%E Edited by _Max Alekseyev_, Jun 16 2010
%E a(19)-a(20) from _Robin Visser_, Sep 30 2023
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