%I #18 May 10 2022 13:58:07
%S 0,1,1,1,0,3,0,5,5,2,6,3,5,0,4,0,0,0,0,2,19,21,2,5,18,8,24,9,12,1,1,
%T 21,0,0,23,20,34,3,28,21,12,10,0,37,33,14,4,9,35,22,14,23,1,0,18,19,
%U 21,4,39,22,16,1,8,6,42,8,16,45,31,50,40,43,18,17,32,38,0,0,26,0,44,1,62,12,4
%N a(n) is the least number k such that 1 <= k < n and prime(n) + 2*prime(n-k) and prime(n) + 2*prime(n+k) are both prime, or 0 if there is no such k.
%C a(n) = 0 for n = 1, 5, 7, 14, 16, 17, 18, 19, 33, 34, 43, 54, 77, 78, 80, 101, 127. Conjecture: these are all the n for which a(n) = 0.
%H Robert Israel, <a href="/A351692/b351692.txt">Table of n, a(n) for n = 1..10000</a>
%e a(6) = 3 because prime(6) + 2*prime(6+3) = 13 + 2*23 = 59 and prime(6) + 2*prime(6-3) = 13 + 2*5 = 23 are prime, while prime(6) + 2*prime(6-1) = 35 is not prime and prime(6) + 2*prime(6+2) = 51 is not prime.
%p N:= 100: # for a(1)..a(N)
%p Primes:= [seq(ithprime(i),i=1..2*N-1)]:
%p f:= proc(k) local p,n;
%p p:= Primes[k];
%p for n from 1 to k-1 do if isprime(p+2*Primes[k+n]) and isprime(p+2*Primes[k-n]) then return n fi
%p od;
%p 0
%p end proc:
%p map(f, [$1..N]);
%o (PARI) a(n) = for (k=1, n-1, my(p=prime(n)); if (isprime(p + 2*prime(n-k)) && isprime(p + 2*prime(n+k)), return(k))); return(0); \\ _Michel Marcus_, May 06 2022
%o (Python)
%o from sympy import isprime, sieve
%o def a(n):
%o pn = sieve[n]
%o for k in range(1, n):
%o if isprime(pn + 2*sieve[n-k]) and isprime(pn + 2*sieve[n+k]):
%o return k
%o return 0
%o print([a(n) for n in range(1, 86)]) # _Michael S. Branicky_, May 10 2022
%Y Cf. A000040, A351693.
%K nonn
%O 1,6
%A _J. M. Bergot_ and _Robert Israel_, May 05 2022