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A140601 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. 2
19, 34, 41, 89, 104, 359, 398, 495, 527, 1845, 2309, 2729, 3539, 4619, 8644, 12923, 14135, 15774 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

Next term (if it exists) is greater than 25325. - Max Alekseyev

For every fixed pair of integers a,b, there exists only a finite number of suitable values of m. - Max Alekseyev

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

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

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)

CROSSREFS

Cf. A140602, A140603.

Sequence in context: A146438 A146571 A235869 * A031206 A214231 A242847

Adjacent sequences:  A140598 A140599 A140600 * A140602 A140603 A140604

KEYWORD

hard,more,nonn

AUTHOR

Andrew V. Sutherland, May 18 2008

EXTENSIONS

Edited by Max Alekseyev, Jun 16 2010

STATUS

approved

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Last modified November 19 07:16 EST 2017. Contains 294916 sequences.