A269671 Integers n such that the concatenation of prime(n) and prime(n+1) and also concatenation of prime(n+1) and prime(n) are prime.

46, 51, 55, 71, 99, 119, 164, 298, 345, 461, 509, 523, 588, 668, 779, 827, 844, 848, 999, 1100, 1151, 1215, 1306, 1321, 1408, 1553, 1568, 1616, 1779, 1900, 1931, 1953, 2102, 2150, 2221, 2444, 2653, 2677, 3116, 3405, 3527, 3731, 3776, 3890, 3898, 3989, 4070, 4188, 4257, 4546, 4556, 4574, 4681, 4694, 4846, 4947, 4948, 4974

Offset: 1

Author

Zak Seidov, Mar 07 2016

Keywords

• nonn
• base

Comments

Difference between prime(n) and prime(n+1) is a multiple of 6, otherwise concatenation prime(n)//prime(n+1) is divisible by 3.

Examples

			prime(46)=199, prime(47)=211 and both 199211 and 211199 are prime,
prime(51)=233, prime(51)=239 and both 233239 and 239233 are prime,
prime(9999972)=179424263, prime(9999973)=179424269 and both 179424263179424269 and 179424269179424263 are prime.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..59542

Crossrefs

Cf. A088712, A088784.

Programs

Mathematica

PrimePi/@Select[Partition[Prime[Range[5000]],2,1],AllTrue[{FromDigits[ Join[ IntegerDigits[ #[[1]]],IntegerDigits[#[[2]]]]],FromDigits[ Join[ IntegerDigits[#[[2]]],IntegerDigits[#[[1]]]]]},PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2021 *)

PARI

isok(n) = {my(sp = Str(prime(n))); my(sq = Str(prime(n+1))); isprime(eval(concat(sp, sq))) && isprime(eval(concat(sq, sp)));} \\ Michel Marcus, Mar 07 2016