%I #35 Mar 02 2018 13:16:27

%S 7,17,23,37,47,59,83,89,107,113,127,131,149,163,173,257,353,433,439,

%T 457,467,521,563,761,773,839,881,953,1009,1031,1213,1307,1319,1321,

%U 1697,1733,1759,1811,1861,1871,1913,1979,2153,2221,2281,2287,2309,2393,2593,2767,2789

%N 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.

%C 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.

%H Robert Israel, <a href="/A295706/b295706.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e Likewise, the difference between the square of 563 and the next prime (569) is 6792, and 6791 and 6793 are twin primes.

%p N:= 10^4: # to get all terms <= N

%p p:= 1: q:= 2: A:= NULL:

%p while p < N do

%p p:= q; q:= nextprime(p);

%p d:= q^2-p^2;

%p if isprime(d+1) and isprime(d-1) then A:= A, p fi

%p od:

%p A; # _Robert Israel_, Mar 02 2018

%t For[p = 1, p < 10000, p++,

%t a = Prime[p];

%t b = Prime[p + 1];

%t c = b^2 - a^2;

%t d = (c + 1);

%t e = (c - 1);

%t If[And[PrimeQ[d] == True, PrimeQ[e] == True], Print[a]];

%t ]

%t (* Second program: *)

%t Select[Partition[Prime@ Range@ 300, 2, 1], AllTrue[{# + 1, # - 1}, PrimeQ] &[#2^2 - #1^2] & @@ # &][[All, 1]] (* _Michael De Vlieger_, Dec 03 2017 *)

%o (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

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

%K nonn

%O 1,1

%A _Geoffrey Marnell_, Nov 25 2017