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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.

%I A178098 #36 May 14 2019 22:03:05
%S A178098 26,30,36,40,42,44,91,95,115,119,133,161,187,247,391,667,1147,1591,
%T A178098 1927,2491,3127,4087,4891,5767,7387,9991,10807,11227,12091,17947,
%U A178098 23707,25591,28891,30967,37627,38407,51067,52891,55687,64507,67591,70747,75067,78391
%N 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.
%C A178098 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
%H A178098 Robert Price, <a href="/A178098/b178098.txt">Table of n, a(n) for n = 1..353</a>
%H A178098 R. J. Mathar, <a href="http://arxiv.org/abs/1109.0922">Corrigendum to "On the divisibility of..."</a>, arXiv:1109.0922 [math.NT], 2011.
%H A178098 V. Shevelev, <a href="http://dx.doi.org/10.1142/S179304210700078X">On divisibility of binomial(n-i-1,i-1) by i</a>, Intl. J. of Number Theory 3, no.1 (2007), 119-139.
%F A178098 {n: A178101(n) = 2}.
%t A178098 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 *)
%o A178098 (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
%Y A178098 Cf. A178101, A178071, A138389, A023201, A178099.
%K A178098 nonn
%O A178098 1,1
%A A178098 _Vladimir Shevelev_, May 20 2010
%E A178098 91 inserted by _R. J. Mathar_, May 28 2010
%E A178098 a(18)-a(36) from _Michel Marcus_, Feb 17 2016
%E A178098 a(37)-a(44) (based on theorem from _Vladimir Shevelev_ in Comments) from _Robert Price_, May 14 2019