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A138763
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.
3
23, 210053, 10480853, 10526459, 11210321, 11722871, 12252263, 12334109, 13647083, 15550331, 23652479, 26724461, 31165133, 48668099, 50599823, 51411989, 56699033, 80672369, 82804763, 90962111, 104066441, 109197401, 109953791, 120560861, 127503113, 153189479, 161933297
OFFSET
1,1
LINKS
EXAMPLE
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.
MATHEMATICA
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];
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, May 11 201
More terms from Amiram Eldar, Jan 01 2020
STATUS
approved