A295706 Primes p for which the difference between p^2 and the square of the next prime is both 1 more and 1 less than a prime.

7, 17, 23, 37, 47, 59, 83, 89, 107, 113, 127, 131, 149, 163, 173, 257, 353, 433, 439, 457, 467, 521, 563, 761, 773, 839, 881, 953, 1009, 1031, 1213, 1307, 1319, 1321, 1697, 1733, 1759, 1811, 1861, 1871, 1913, 1979, 2153, 2221, 2281, 2287, 2309, 2393, 2593, 2767, 2789

Offset

1,1

Comments

I.e., primes p for which the difference between p^2 and the square of the next prime is the average of a twin prime pair.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The primes 7 and 11 are consecutive and their squares are 49 and 121. The difference is 72, and both 71 and 73 are prime.
Likewise, the difference between the square of 563 and the next prime (569) is 6792, and 6791 and 6793 are twin primes.

Maple

N:= 10^4: # to get all terms <= N
p:= 1: q:= 2: A:= NULL:
while p < N do
  p:= q; q:= nextprime(p);
  d:= q^2-p^2;
  if isprime(d+1) and isprime(d-1) then A:= A, p fi
od:
A; # Robert Israel, Mar 02 2018

Mathematica

For[p = 1, p < 10000, p++,
a = Prime[p];
b = Prime[p + 1];
c = b^2 - a^2;
d = (c + 1);
e = (c - 1);
If[And[PrimeQ[d] == True, PrimeQ[e] == True], Print[a]];
]
(* Second program: *)
Select[Partition[Prime@ Range@ 300, 2, 1], AllTrue[{# + 1, # - 1}, PrimeQ] &[#2^2 - #1^2] & @@ # &][[All, 1]] (* Michael De Vlieger, Dec 03 2017 *)

Prog

(PARI) lista(nn) = { my(pp=2); forprime(p=3, nn, my(d=p^2-pp^2); if(isprime(d+1) && isprime(d-1), print1(pp, ", ")); pp=p); } \\ Iain Fox, Dec 03 2017

Crossrefs

Cf. A014574 (average of twin prime pairs), A069482 (difference between squares of consecutive primes).

Sequence in context: A088546 A265816 A246717 * A265792 A322669 A115591

Adjacent sequences: A295703 A295704 A295705 * A295707 A295708 A295709

Keyword

nonn

Author

Geoffrey Marnell, Nov 25 2017

Status

approved