A235806 Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 - prime((p + 1)/2) both prime.

7, 11, 19, 29, 41, 43, 53, 59, 89, 109, 139, 179, 181, 229, 379, 401, 421, 431, 541, 587, 659, 811, 991, 1069, 1103, 1117, 1231, 1259, 1459, 1471, 1619, 1709, 1831, 1951, 2179, 2791, 2797, 2833, 2851, 3001, 3391, 3571, 3617, 3631, 3637, 3671, 3793, 3863, 3929, 3967

Offset

1,1

Comments

By the conjecture in A235805, this sequence should have infinitely many terms.

Links

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

Example

a(1) = 7 since neither (3^2-1)/4 - prime((3-1)/2) = 0 nor (5^2-1)/4 - prime((5+1)/2) = 1 is prime, but (7^2-1)/4 - prime((7-1)/2) = 12 - 5 = 7 and (7^2-1)/4 - prime((7+1)/2) = 12 - 7 = 5 are both prime.

Mathematica

q[n_]:=PrimeQ[n(n+1)-Prime[n]]&&PrimeQ[n(n+1)-Prime[n+1]]
n=0; Do[If[q[(Prime[k]-1)/2], n=n+1; Print[n, " ", Prime[k]]], {k, 2, 1000}]
Select[Prime[Range[2, 600]], AllTrue[(#^2-1)/4-{Prime[(#-1)/2], Prime[ (#+1)/2]}, PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2020 *)

Crossrefs

Cf. A000040, A235592, A235727, A235805.

Sequence in context: A192187 A053403 A032672 * A238501 A133425 A103802

Adjacent sequences: A235803 A235804 A235805 * A235807 A235808 A235809

Keyword

nonn

Author

Zhi-Wei Sun, Jan 16 2014

Status

approved