A327755 - OEIS

%I #23 Mar 12 2024 22:50:45

%S 1,3,5,7,9,11,13,17,19,21,23,25,27,29,31,37,41,43,47,49,53,55,57,59,

%T 61,65,67,71,73,75,79,81,83,89,97,101,103,107,109,113,115,119,121,125,

%U 127,131,133,137,139,149,151,157,163,167,169,171,173,179,181,183,191,193,197,199

%N Odd integers k such that binomial(k-1,(k-1)/2) is coprime to k.

%C Integer k is in this sequence if for every prime p|k, the base-p representation of (k-1)/2 is composed of digits not exceeding (p-1)/2.

%C Contains all odd primes. Composite terms are listed in A327756.

%H David A. Corneth, <a href="/A327755/b327755.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms of Harvey P. Dale)

%e 21 is in the sequence because it is odd and binomial(20 - 1, (20 - 1)/2) = 2^2 * 11 * 13 * 17 * 19 which is coprime 3 * 7. - _David A. Corneth_, Mar 09 2024

%t Select[Range[1,401,2],CoprimeQ[#,Binomial[#-1,(#-1)/2]]&] (* _Harvey P. Dale_, Feb 25 2024 *)

%o (PARI) is(n) = {

%o 	if(!bitand(n, 1), return(0));

%o 	my(f = factor(n), h = (n-1)/2, v);

%o 	for(i = 1, #f~,

%o 		v = val(n-1, f[i,1]) - 2*val(h, f[i,1]);	

%o 		if(v > 0,

%o 			return(0)

%o 		)

%o 	); 1

%o }

%o val(n, p) = my(r=0); while(n, r+=n\=p); r \\ _David A. Corneth_, Mar 09 2024

%Y Cf. A327756.

%K nonn

%O 1,2

%A _Max Alekseyev_, Sep 24 2019