1,1

This sequence was first suggested by Stefan Steinerberger, who conjectures that there are infinitely many terms.

It is known that a(19) > 18750 (by exhaustive search).

For all known entries (up to a(18)), 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.

Open questions: (1) Must the pair (a,b) for a given a(n) be unique? (2) Does every solution have b-a <= 2?

The following is a complete list of solutions with a(n) <= 18750

C(19,3)+C(19,5) divides C(19,8)

C(34,6)+C(34,7) divides C(34,13)

C(41,5)+C(41,7) divides C(41,12)

C(89,7)+C(89,8) divides C(89,15)

C(104,3)+C(104,4) divides C(104,7)

C(359,5)+C(359,6) divides C(359,11)

C(398,20)+C(398,21) divides C(398,41)

C(495,12)+C(495,14) divides C(495,26)

C(527,7)+C(527,9) divides C(527,16)

C(1845,15)+C(1845,17) divides C(1845,32)

C(2309,5)+C(2309,6) divides C(2309,11)

C(2729,19)+C(2729,20) divides C(2729,39)

C(3539,35)+C(3539,36) divides C(3539,71)

C(4619,11)+C(4619,12) divides C(4619,23)

C(8644,18)+C(8644,19) divides C(8644,37)

C(12923,34)+C(12923,36) divides C(12923,70)

C(14135,30)+C(14135,31) divides C(14135,61)

C(15774,24)+C(15774,26) divides C(15774,50)

a(1) = 19 because C(19,3) + C(19,5) = 969 + 11628 = 12597 divides C(19,8) = 75582, and 19 is the first nonnegative integer for which such a condition holds.

Cf. A140602, A140603.

Sequence in context: A061962 A116168 A106527 this_sequence A031206 A044064 A044445

Adjacent sequences: A140598 A140599 A140600 this_sequence A140602 A140603 A140604

hard,more,nonn

Andrew V. Sutherland (drew(AT)math.mit.edu), May 18 2008