A178071 - OEIS

%I #44 Aug 07 2022 02:07:49

%S 14,16,18,22,27,28,39,55,65,77,85,221,437,1517,2021,4757,6557,9797,

%T 11021,12317,16637,27221,38021,50621,53357,77837,95477,99221,123197,

%U 145157,159197,194477,210677,216221,239117,250997,378221,416021,455621,549077,576077

%N 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 Note that every d > 1 divides binomial(k-d-1, d-1), if gcd(k,d)=1.

%C 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 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 Robert Price, <a href="/A178071/b178071.txt">Table of n, a(n) for n = 1..179</a>

%H 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 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 {k: A178101(k) = 1}.

%p filter:= proc(n) local d, b,count;

%p   count:= 0;

%p   b:= 1;

%p   for d from 2 to n/2 do

%p      b:= b * (n-2*d+1)*(n-2*d+2)/(n-d)/(d-1);

%p      if igcd(d,n) <> 1 and b mod d = 0 then

%p         count:= count+1;

%p         if count = 2 then return false fi;

%p      fi

%p   od;

%p   evalb(count=1);

%p end proc:

%p select(filter, [$1..10^4]); # _Robert Israel_, Feb 17 2016

%t 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 (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 Cf. A001359, A138389, A178101.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, May 19 2010

%E a(15)-a(23) from _Michel Marcus_, Feb 17 2016

%E a(24)-a(41) (from theorem in the Shevelev link) from _Robert Price_, May 14 2019