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A178071 Numbers k such that exactly one d, 2 <= d <= k/2, exists which divides binomial(k-d-1, d-1) and is not coprime to k.

%I A178071 #44 Aug 07 2022 02:07:49
%S A178071 14,16,18,22,27,28,39,55,65,77,85,221,437,1517,2021,4757,6557,9797,
%T A178071 11021,12317,16637,27221,38021,50621,53357,77837,95477,99221,123197,
%U A178071 145157,159197,194477,210677,216221,239117,250997,378221,416021,455621,549077,576077
%N A178071 Numbers k such that exactly one d, 2 <= d <= k/2, exists which divides binomial(k-d-1, d-1) and is not coprime to k.
%C A178071 Note that every d > 1 divides binomial(k-d-1, d-1), if gcd(k,d)=1.
%C A178071 As shown in the Shevelev link, the sequence contains p*(p+4) for every p >= 7 in A023200.  Thus it is infinite if A023200 is infinite. - _Robert Israel_, Feb 18 2016
%C A178071 Moreover, similar to proof of Theorem 1 in this link, one can prove that a number m > 85 is a member if and only if it has such a form. - _Vladimir Shevelev_, Feb 23 2016
%H A178071 Robert Price, <a href="/A178071/b178071.txt">Table of n, a(n) for n = 1..179</a>
%H A178071 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 A178071 Vladimir 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 A178071 {k: A178101(k) = 1}.
%p A178071 filter:= proc(n) local d, b,count;
%p A178071   count:= 0;
%p A178071   b:= 1;
%p A178071   for d from 2 to n/2 do
%p A178071      b:= b * (n-2*d+1)*(n-2*d+2)/(n-d)/(d-1);
%p A178071      if igcd(d,n) <> 1 and b mod d = 0 then
%p A178071         count:= count+1;
%p A178071         if count = 2 then return false fi;
%p A178071      fi
%p A178071   od;
%p A178071   evalb(count=1);
%p A178071 end proc:
%p A178071 select(filter, [$1..10^4]); # _Robert Israel_, Feb 17 2016
%t A178071 Select[Range@ 4000, Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 1]] (* _Michael De Vlieger_, Feb 17 2016 *)
%o A178071 (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 > 1, return (0));); nb == 1;} \\ _Michel Marcus_, Feb 17 2016
%Y A178071 Cf. A001359, A138389, A178101.
%K A178071 nonn
%O A178071 1,1
%A A178071 _Vladimir Shevelev_, May 19 2010
%E A178071 a(15)-a(23) from _Michel Marcus_, Feb 17 2016
%E A178071 a(24)-a(41) (from theorem in the Shevelev link) from _Robert Price_, May 14 2019