

A236552


a(n) = {0 < k < n: 6*k  1, 6*k + 1, 6*k + 5 and prime(nk) + 6 are all prime}.


1



0, 0, 0, 1, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 4, 5, 4, 4, 3, 4, 3, 5, 5, 5, 2, 2, 4, 3, 4, 3, 3, 4, 5, 3, 5, 4, 6, 6, 6, 4, 4, 1, 3, 6, 5, 2, 4, 4, 5, 6, 8, 6, 5, 4, 8, 4, 4, 5, 7, 6, 4, 4, 4, 6, 6, 5, 5, 6, 6, 7, 6, 4, 3, 5, 7, 3, 4, 4, 3, 6, 5, 5, 6, 6, 7, 5, 4, 3, 5, 5, 3, 3, 2, 2
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OFFSET

1,5


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 4, there is a positive integer k < n such that 6*k  5, 6*k  1, 6*k + 1 and prime(nk) + 6 are all prime.
(iii) Any integer n > 7 can be written as p + q with q > 0 such that p, p + 6 and prime(q) + 6 are all prime.
(iv) Each integer n > 4 can be written as k*(k+1) + m with k > 0 and m > 0 such that prime(m) + 6 is prime.


LINKS



EXAMPLE

a(4) = 1 since 6*1  1, 6*1 + 1, 6*1 + 5 and prime(41) + 6 = 11 are all prime.
a(48) = 1 since 6*32  1 = 191, 6*32 + 1 = 193, 6*32 + 5 = 197 and prime(4832) + 6 = 53 + 6 = 59 are all prime.


MATHEMATICA

p[n_]:=PrimeQ[6n1]&&PrimeQ[6n+1]&&PrimeQ[6n+5]
q[n_]:=PrimeQ[Prime[n]+6]
a[n_]:=Sum[If[p[k]&&q[nk], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



