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A238756 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k)) - prime(k) + 1 and prime(prime(m)) - prime(m) + 1 are all prime. 3
0, 1, 2, 3, 3, 2, 3, 3, 3, 4, 2, 5, 4, 3, 6, 4, 4, 3, 3, 6, 5, 5, 4, 6, 6, 5, 6, 2, 7, 5, 5, 6, 4, 4, 4, 5, 5, 8, 2, 5, 4, 5, 8, 2, 5, 2, 7, 4, 8, 6, 4, 5, 3, 8, 4, 7, 5, 3, 7, 7, 5, 7, 5, 7, 9, 8, 7, 5, 9, 7, 10, 9, 7, 7, 6, 9, 10, 4, 5, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Conjecture: a(n) > 0 for all n > 1.

We have verified this for n up to 10^7.

The conjecture suggests that there are infinitely many primes p with 2*pi(p) + 1 and prime(p) - p + 1 both prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

EXAMPLE

a(6) = 2 since 6 = 2 + 4 with 2*2 + 1 = 5, prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 3 and prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 all prime, and 6 = 3 + 3 with 2*3 + 1 = 7 and prime(prime(3)) - prime(3) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 both prime.

MATHEMATICA

p[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]

a[n_]:=Sum[If[PrimeQ[2k+1]&&p[k]&&p[n-k], 1, 0], {k, 1, n-1}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A218829, A234694, A234695, A235189, A236832, A237413, A238134.

Sequence in context: A276859 A007538 A242285 * A025076 A110006 A289831

Adjacent sequences:  A238753 A238754 A238755 * A238757 A238758 A238759

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 05 2014

STATUS

approved

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Last modified October 22 12:30 EDT 2019. Contains 328318 sequences. (Running on oeis4.)