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

%I A268082 #27 Sep 08 2022 08:46:15
%S A268082 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,
%T A268082 53,55,59,61,64,67,71,73,79,81,83,89,93,97,101,103,107,109,111,113,
%U A268082 119,121,125,127,128,131,137,139,149,151,155,157,161,163,167,169,173,179
%N A268082 Numbers n such that gcd(binomial(2*n-1,n), n) is equal to 1.
%C A268082 Or numbers n such that A088218(n) is coprime to n.
%C A268082 The power of primes (A000961) are terms of this sequence.
%C A268082 From _Robert Israel_, Jan 26 2016: (Start)
%C A268082 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.
%C A268082 The only even terms are powers of 2.
%C A268082 All terms divisible by 3 are in A005836, and all terms divisible by 5 are in A037453. (End)
%C A268082 A082916 (after 0) lists the odd terms of this sequence. - _Bruno Berselli_, Jan 26 2015
%H A268082 Robert Israel, <a href="/A268082/b268082.txt">Table of n, a(n) for n = 1..10000</a>
%H A268082 Victor J.W. Guo and Jiang Zeng, <a href="http://dx.doi.org/10.1016/j.jnt.2009.07.005">Factors of binomial sums from the Catalan triangle</a>, Journal of Number Theory 130 (2010) 172-186.
%H A268082 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kummer%27s_theorem">Kummer's theorem</a>
%H A268082 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas%27_theorem">Lucas's theorem</a>
%e A268082 For n=3, binomial(2*n-1, n) = binomial(5, 3) = 10 and 10 is coprime to 3, so 3 is in the sequence.
%p A268082 filter:= proc(n) local F,p;
%p A268082 if n::even then evalb(n = 2^padic:-ordp(n,2))
%p A268082 else
%p A268082    F:= numtheory:-factorset(n);
%p A268082    for p in F do
%p A268082      if max(convert(n,base,p)) > p/2 then return false fi;
%p A268082    od;
%p A268082    true
%p A268082 fi
%p A268082 end proc:
%p A268082 select(filter, [$1..1000]); # _Robert Israel_, Jan 26 2016
%t A268082 Select[Range@ 180, GCD[Binomial[2 # - 1, #], #] == 1 &] (* _Michael De Vlieger_, Jan 26 2016 *)
%o A268082 (PARI) isok(n) = gcd(binomial(2*n-1,n), n) == 1;
%o A268082 (PARI) lista(nn) = for(n=1, nn, if(gcd(binomial(2*n-1, n), n) == 1, print1(n, ", "))); \\ _Altug Alkan_, Jan 26 2016
%o A268082 (Magma) [n: n in [1..200] | Gcd(Binomial(2*n-1,n), n) eq 1]; // _Vincenzo Librandi_, Jan 26 2016
%Y A268082 Cf. A000961, A005836, A037453, A082916 , A088218, A268083.
%K A268082 nonn
%O A268082 1,2
%A A268082 _Michel Marcus_, Jan 26 2016