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

Author

Vladimir Shevelev, May 20 2010

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

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

Formula

{n: A178101(n) = 2}.

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