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.

14, 16, 18, 22, 27, 28, 39, 55, 65, 77, 85, 221, 437, 1517, 2021, 4757, 6557, 9797, 11021, 12317, 16637, 27221, 38021, 50621, 53357, 77837, 95477, 99221, 123197, 145157, 159197, 194477, 210677, 216221, 239117, 250997, 378221, 416021, 455621, 549077, 576077

Offset: 1

Author

Vladimir Shevelev, May 19 2010

Keywords

• nonn

Comments

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

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

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

Links

• Robert Price, Table of n, a(n) for n = 1..179
• R. J. Mathar, Corrigendum to "On the divisibility of...", arXiv:1109.0922 [math.NT], 2011.
• Vladimir 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. A001359, A138389, A178101.

Programs

Maple

filter:= proc(n) local d, b,count;
  count:= 0;
  b:= 1;
  for d from 2 to n/2 do
     b:= b * (n-2*d+1)*(n-2*d+2)/(n-d)/(d-1);
     if igcd(d,n) <> 1 and b mod d = 0 then
        count:= count+1;
        if count = 2 then return false fi;
     fi
  od;
  evalb(count=1);
end proc:
select(filter, [$1..10^4]); # Robert Israel, Feb 17 2016

Mathematica

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 *)

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

Formula

{k: A178101(k) = 1}.

Extensions

a(15)-a(23) from Michel Marcus, Feb 17 2016

a(24)-a(41) (from theorem in the Shevelev link) from Robert Price, May 14 2019