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%I #16 Jan 30 2024 08:23:40
%S 0,0,0,1,0,1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,1,1,0,1,0,1,1,1,1,0,1,
%T 0,1,1,1,1,1,0,1,0,1,0,1,1,1,0,1,1,0,1,1,1,1,0,1,1,0,1,1,1,0,1,1,1,1,
%U 0,1,1,1,1,1,1,1,1,1,1,1,0,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,0,1
%N a(n)=1 if there is a number of the form 6k+3 such that prime(n) < 6k+3 < prime(n+1), otherwise 0.
%C Except for first two primes and twin primes, there is always at least one number of the form 6k+3 between two successive primes.
%H Karl-Heinz Hofmann, <a href="/A106002/b106002.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3)=0 because between prime(3)=5 and prime(4)=7 there are no numbers of the form 6k+3;
%e a(4)=1 because between prime(4)=7 and prime(5)=11 there is 9=6*1+3.
%t Table[If[Prime[n]<6Ceiling[Prime[n]/6]+3<Prime[n+1] || Prime[n]<6Floor[Prime[n]/6]+3<Prime[n+1] ,1,0],{n,105}] (* _James C. McMahon_, Jan 29 2024 *)
%o (PARI) a(n) = my(p=prime(n)); for(k=p+1, nextprime(p+1)-1, if (!((k-3) % 6), return(1))); \\ _Michel Marcus_, Jan 30 2024
%o (Python)
%o from sympy import sieve
%o def A106002(n):
%o for comp in range(sieve[n]+1, sieve[n+1]):
%o if (comp-3) % 6 == 0: return 1
%o return 0 # _Karl-Heinz Hofmann_, Jan 30 2024
%Y Same as A100810 after first term.
%K easy,nonn
%O 1,1
%A _Giovanni Teofilatto_, Apr 29 2005
%E Edited by _Ray Chandler_, Oct 17 2006
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