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
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