A225575 Primes p such that if q is the next prime after p then the concatenation of p with q and the concatenation of q with p are both primes.

199, 233, 257, 353, 523, 653, 971, 1973, 2333, 3259, 3637, 3761, 4283, 4993, 5927, 6353, 6529, 6563, 7907, 8831, 9293, 9851, 10711, 10861, 11731, 13037, 13177, 13681, 15241, 16381, 16693, 16931, 18341, 18899, 19577, 21787, 23857, 24071, 28621, 31657, 32911

Offset: 1

Author

W. Edwin Clark, May 10 2013

Keywords

• nonn
• base

Comments

This sequence was suggested by Puzzle 687 at Prime Puzzles (see Links).

If q = p + 2 (that is, p is the lesser of a twin prime, A001359), then p is not in the sequence. - Alonso del Arte, May 10 2013

In general, q-p == 0 (mod 6). - Zak Seidov, May 11 2013

Examples

			The prime following 199 is 211 and both 199211 and 211199 are prime.
		

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..10000 (terms 1..250 from Paolo P. Lava)
• Carlos Rivera, Puzzle 687: 17769643, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A030459.

Programs

Magma

conc:=func; [p:p in PrimesUpTo(17000)| IsPrime(conc(p,(NextPrime(p)))) and IsPrime(conc(NextPrime(p),p))]; // Marius A. Burtea, Jan 25 2020

Maple

a:=NULL;
for i from 1 to 2000 do
  p:=ithprime(i);
  q:=nextprime(p);
  s:=convert(p,string);
  t:=convert(q,string);
  if isprime(parse(cat(s,t))) and isprime(parse(cat(t,s))) then a:=a,p; fi;
od:
a;

Mathematica

concatQ[{a_,b_}]:=Module[{idna=IntegerDigits[a],idnb=IntegerDigits[b]},AllTrue[ FromDigits/@ {Join[idna,idnb], Join[idnb,idna]},PrimeQ]]; Transpose[Select[ Partition[ Prime[Range[2000]],2,1],concatQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 16 2015 *)