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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. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 20:01 EDT 2019. Contains 328373 sequences. (Running on oeis4.)