%I #16 Sep 08 2022 08:45:33

%S 23,210053,10480853,10526459,11210321,11722871,12252263,12334109,

%T 13647083,15550331,23652479,26724461,31165133,48668099,50599823,

%U 51411989,56699033,80672369,82804763,90962111,104066441,109197401,109953791,120560861,127503113,153189479,161933297

%N Primes p1 such that p1^2 + p2^3 and p1^3 + p2^2 are averages of twin primes. p1 and p2 are consecutive primes, p1 < p2.

%H Amiram Eldar, <a href="/A138763/b138763.txt">Table of n, a(n) for n = 1..120</a>

%e p1 = 23 is a term since the next prime is p2 = 29, and both p1^2 + p2^3 = 24918 and p1^3 + p2^2 = 13008 are averages of twin primes.

%t a={};Do[p1=Prime[n];p2=Prime[n+1];p3=p1^2+p2^3;p4=p1^3+p2^2;If[PrimeQ[p3-1]&&PrimeQ[p3+1]&&PrimeQ[p4-1]&&PrimeQ[p4+1],AppendTo[a,p1]],{n,13^5}];Print[a];

%t cpQ[{a_,b_}]:=Module[{p3=a^2+b^3,p4=a^3+b^2},AllTrue[{p3+1,p3-1, p4+1, p4-1}, PrimeQ]]; Transpose[Select[Partition[Prime[ Range[ 2*10^6]], 2,1],cpQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, May 11 2015 *)

%o (Magma) [p:p in PrimesInInterval(1,11000000)| IsPrime(a+1) and IsPrime(a-1) and IsPrime(b+1) and IsPrime(b-1) where a is p^2+q^3 where b is p^3+q^2 where q is NextPrime(p)]; // _Marius A. Burtea_, Jan 01 2020

%Y Cf. A001097, A001359, A006512, A014574.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, May 15 2008

%E More terms from _Harvey P. Dale_, May 11 201

%E More terms from _Amiram Eldar_, Jan 01 2020