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%I A295706 #35 Mar 02 2018 13:16:27
%S A295706 7,17,23,37,47,59,83,89,107,113,127,131,149,163,173,257,353,433,439,
%T A295706 457,467,521,563,761,773,839,881,953,1009,1031,1213,1307,1319,1321,
%U A295706 1697,1733,1759,1811,1861,1871,1913,1979,2153,2221,2281,2287,2309,2393,2593,2767,2789
%N 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.
%C A295706 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 A295706 Robert Israel, <a href="/A295706/b295706.txt">Table of n, a(n) for n = 1..10000</a>
%e A295706 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 A295706 Likewise, the difference between the square of 563 and the next prime (569) is 6792, and 6791 and 6793 are twin primes.
%p A295706 N:= 10^4: # to get all terms <= N
%p A295706 p:= 1: q:= 2: A:= NULL:
%p A295706 while p < N do
%p A295706   p:= q; q:= nextprime(p);
%p A295706   d:= q^2-p^2;
%p A295706   if isprime(d+1) and isprime(d-1) then A:= A, p fi
%p A295706 od:
%p A295706 A; # _Robert Israel_, Mar 02 2018
%t A295706 For[p = 1, p < 10000, p++,
%t A295706 a = Prime[p];
%t A295706 b = Prime[p + 1];
%t A295706 c = b^2 - a^2;
%t A295706 d = (c + 1);
%t A295706 e = (c - 1);
%t A295706 If[And[PrimeQ[d] == True, PrimeQ[e] == True], Print[a]];
%t A295706 ]
%t A295706 (* Second program: *)
%t A295706 Select[Partition[Prime@ Range@ 300, 2, 1], AllTrue[{# + 1, # - 1}, PrimeQ] &[#2^2 - #1^2] & @@ # &][[All, 1]] (* _Michael De Vlieger_, Dec 03 2017 *)
%o A295706 (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 A295706 Cf. A014574 (average of twin prime pairs), A069482 (difference between squares of consecutive primes).
%K A295706 nonn
%O A295706 1,1
%A A295706 _Geoffrey Marnell_, Nov 25 2017

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