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.

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

Amiram Eldar, Table of n, a(n) for n = 1..120

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

Cf. A001097, A001359, A006512, A014574.

Sequence in context: A101699 A122148 A068736 * A156176 A013772 A320442

Adjacent sequences: A138760 A138761 A138762 * A138764 A138765 A138766

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, May 15 2008

Extensions

More terms from Harvey P. Dale, May 11 201

More terms from Amiram Eldar, Jan 01 2020

Status

approved