A235681 Primes p with prime(p) - p + 1 and p*(p+1) - prime(p) both prime.

2, 3, 5, 41, 61, 71, 89, 271, 281, 293, 337, 499, 571, 751, 907, 911, 1093, 1531, 2027, 2341, 2707, 2861, 3011, 3359, 3391, 3511, 4133, 5179, 5189, 5483, 5573, 5657, 5867, 6577, 6827, 7159, 7411, 7753, 7879, 8179, 8467, 9209, 9391, 9419, 9433, 10259, 10303, 10859, 10993, 11287

Offset

1,1

Comments

This is the intersection of A234695 and A235661. For any prime p in this sequence, p^2 + 1 is the sum of the two primes prime(p) - p + 1 and p*(p+1) - prime(p).

By the conjecture in A235682, this sequence should have infinitely many terms.

Links

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

Example

a(1) = 2 since prime(2) - 2 + 1 = 2 and 2*3 - prime(2) = 3 are both prime.
a(2) = 3 since prime(3) - 3 + 1 = 3 and 3*4 - prime(3) = 7 are both prime.
a(3) = 5 since prime(5) - 5 + 1 = 7 and 5*6 - prime(5) = 19 are both prime.

Mathematica

PQ[n_]:=PrimeQ[Prime[n]-n+1]&&PrimeQ[n(n+1)-Prime[n]]
n=0; Do[If[PQ[Prime[k]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 1000}]

Crossrefs

Cf. A000040, A234694, A234695, A232353, A235189, A235330, A235613, A235614, A235661, A235682.

Sequence in context: A215309 A215105 A088483 * A322748 A224781 A136015

Adjacent sequences: A235678 A235679 A235680 * A235682 A235683 A235684

Keyword

nonn

Author

Zhi-Wei Sun, Jan 13 2014

Status

approved