A178098 Numbers n such that exactly two positive d in the range d <= n/2 exist which divide binomial(n-d-1, d-1) and which are not coprime to n.

26, 30, 36, 40, 42, 44, 91, 95, 115, 119, 133, 161, 187, 247, 391, 667, 1147, 1591, 1927, 2491, 3127, 4087, 4891, 5767, 7387, 9991, 10807, 11227, 12091, 17947, 23707, 25591, 28891, 30967, 37627, 38407, 51067, 52891, 55687, 64507, 67591, 70747, 75067, 78391

Offset

1,1

Comments

Theorem: A number m > 161 is a member if and only if it is a product p*(p+6) such that both p and p+6 are primes (A023201). The proof is similar to that of Theorem 1 in the Shevelev link. - Vladimir Shevelev, Feb 23 2016

Links

Robert Price, Table of n, a(n) for n = 1..353

R. J. Mathar, Corrigendum to "On the divisibility of...", arXiv:1109.0922 [math.NT], 2011.

V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory 3, no.1 (2007), 119-139.

Formula

{n: A178101(n) = 2}.

Mathematica

Select[Range@ 4000, Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 2]] (* Michael De Vlieger, Feb 17 2016 *)

Prog

(PARI) isok(n)=my(nb = 0); for (d=2, n\2, if ((gcd(d, n) != 1) && ((binomial(n-d-1, d-1) % d) == 0), nb++); if (nb > 2, return (0)); ); nb == 2; \\ Michel Marcus, Feb 17 2016

Crossrefs

Cf. A178101, A178071, A138389, A023201, A178099.

Sequence in context: A316617 A303815 A069962 * A278779 A045163 A282110

Adjacent sequences: A178095 A178096 A178097 * A178099 A178100 A178101

Keyword

nonn

Author

Vladimir Shevelev, May 20 2010

Extensions

91 inserted by R. J. Mathar, May 28 2010

a(18)-a(36) from Michel Marcus, Feb 17 2016

a(37)-a(44) (based on theorem from Vladimir Shevelev in Comments) from Robert Price, May 14 2019

Status

approved