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A377014
a(n) is the number of primes p such that p - 6, p + 6 and 2*n - p are also primes.
0
0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 2, 3, 4, 1, 3, 3, 0, 4, 4, 2, 2, 3, 3, 3, 6, 3, 4, 6, 0, 5, 5, 1, 6, 4, 3, 5, 6, 4, 3, 9, 3, 2, 8, 2, 4, 7, 2, 4, 3, 3, 5, 5, 6, 4, 9, 4, 4, 11, 2, 5, 10, 1, 4, 4, 4, 4, 4, 5, 2, 7, 4, 4, 9, 2, 5, 6, 0, 6, 7, 5, 3, 6, 5, 1, 10, 7, 4, 9, 2, 5, 9, 2, 6, 5, 4, 5, 4, 4
OFFSET
1,8
COMMENTS
Conjecture: a(n) = 0 only when n = 1, 2, 3, 4, 5, 6, 19, 31, 331, 499.
EXAMPLE
a(7) = 1 since only when p = 11 are p - 6, p + 6 and 2n - p all prime.
a(12) = 3 from the cases when p is 11, 13 or 17:
when p = 11, {p - 6, p + 6, 2n - p} = {5, 17, 13} are all prime;
when p = 13, {p - 6, p + 6, 2n - p} = {7, 13, 19, 11} are all prime;
when p = 17, {p - 6, p + 6, 2n - p} = {11, 17, 23, 7} are all prime.
a(19) = 0 since 2n = 38 = 7 + 31 = 19 + 19 = 31 + 7, and none of p = 7, 19, 31 can make p - 6 and p + 6 both prime.
MAPLE
f:= proc(n) local i;
nops(select(p -> andmap(isprime, [p, p-6, p+6, 2*n-p]), [seq(i, i=3..2*n, 2)]))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024
MATHEMATICA
m = 200; ps = {}; p = 7; While[p = NextPrime[p]; If[PrimeQ[p - 6] && PrimeQ[p + 6], AppendTo[ps, p]]; p < 2*m]; a = {}; Do[ct = 0; k = 0; While[k++; ps[[k]] < n, q = n - ps[[k]]; If[PrimeQ[q], ct++]]; AppendTo[a, ct]; If[ct == 0, AppendTo[b, n]], {n, 2, m, 2}]; a
CROSSREFS
Sequence in context: A029233 A147981 A051888 * A327966 A366580 A088019
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Oct 12 2024
STATUS
approved