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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.
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%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