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A235806
Odd primes p with (p^2 - 1)/4 - prime((p - 1)/2) and (p^2 - 1)/4 - prime((p + 1)/2) both prime.
3
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.
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 16 2014
STATUS
approved