A268082 Numbers n such that gcd(binomial(2*n-1,n), n) is equal to 1.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 93, 97, 101, 103, 107, 109, 111, 113, 119, 121, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179

Offset

1,2

Comments

Or numbers n such that A088218(n) is coprime to n.

The power of primes (A000961) are terms of this sequence.

From Robert Israel, Jan 26 2016: (Start)

By Lucas's theorem, these are the n such that for every prime p dividing n, no base-p digit of n is greater than the corresponding base-p digit of 2n-1. Equivalently (Kummer's theorem), there are no carries in base-p addition of n and n-1. Thus if p is odd, each base-p digit of n is less than p/2.

The only even terms are powers of 2.

All terms divisible by 3 are in A005836, and all terms divisible by 5 are in A037453. (End)

A082916 (after 0) lists the odd terms of this sequence. - Bruno Berselli, Jan 26 2015

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Victor J.W. Guo and Jiang Zeng, Factors of binomial sums from the Catalan triangle, Journal of Number Theory 130 (2010) 172-186.

Wikipedia, Kummer's theorem

Wikipedia, Lucas's theorem

Example

For n=3, binomial(2*n-1, n) = binomial(5, 3) = 10 and 10 is coprime to 3, so 3 is in the sequence.

Maple

filter:= proc(n) local F, p;
if n::even then evalb(n = 2^padic:-ordp(n, 2))
else
   F:= numtheory:-factorset(n);
   for p in F do
     if max(convert(n, base, p)) > p/2 then return false fi;
   od;
   true
fi
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 26 2016

Mathematica

Select[Range@ 180, GCD[Binomial[2 # - 1, #], #] == 1 &] (* Michael De Vlieger, Jan 26 2016 *)

Prog

(PARI) isok(n) = gcd(binomial(2*n-1, n), n) == 1;
(PARI) lista(nn) = for(n=1, nn, if(gcd(binomial(2*n-1, n), n) == 1, print1(n, ", "))); \\ Altug Alkan, Jan 26 2016
(Magma) [n: n in [1..200] | Gcd(Binomial(2*n-1, n), n) eq 1]; // Vincenzo Librandi, Jan 26 2016

Crossrefs

Cf. A000961, A005836, A037453, A082916 , A088218, A268083.

Sequence in context: A323306 A325247 A306013 * A247199 A087441 A326645

Adjacent sequences: A268079 A268080 A268081 * A268083 A268084 A268085

Keyword

nonn

Author

Michel Marcus, Jan 26 2016

Status

approved