A268083 -id:A268083 - cp's OEIS frontend

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

Michel Marcus, Jan 26 2016

nonn

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

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

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

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

[n: n in [1..200] | Gcd(Binomial(2*n-1,n), n) eq 1]; // Vincenzo Librandi, Jan 26 2016

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

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

isok(n) = gcd(binomial(2*n-1,n), n) == 1;

lista(nn) = for(n=1, nn, if(gcd(binomial(2*n-1, n), n) == 1, print1(n, ", "))); \\ Altug Alkan, Jan 26 2016

cp's OEIS Frontend

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

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