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

1,1

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.

Formula

{k: A178101(k) = 1}.

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 *)

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 > 1, return (0)); ); nb == 1; } \\ Michel Marcus, Feb 17 2016

Crossrefs

Cf. A001359, A138389, A178101.

Sequence in context: A241178 A158282 A053425 * A177982 A034305 A091898

Adjacent sequences: A178068 A178069 A178070 * A178072 A178073 A178074

Keyword

nonn

Author

Vladimir Shevelev, May 19 2010

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

Status

approved