A383810 Primes which satisfy the requirements of A380943 in more than one way.

373, 1913, 3733, 6737, 7937, 11353, 13997, 19937, 19997, 23773, 24113, 29347, 31181, 31193, 31907, 34729, 37277, 38237, 41593, 47293, 59929, 71971, 72719, 73823, 74177, 79337, 79613, 82373, 83773, 83911, 88397, 100913, 103997

Offset: 1

Author

James C. McMahon and Robert G. Wilson v, May 11 2025

Keywords

• base
• nonn

Comments

The requirements of A380943 are that primes, written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

The number of terms <= 10^k beginning with k=1: 0, 0, 1, 5, 31, 285, 930, 5625, 28137, 205416, ....

Examples

			373 is a member since 373 is the 74th prime, p=3 and q=73, and the reverse concatenation is 733 which is the 130th prime. In another way, p=37 and q=3 and the reverse concatenation is 337, the 68th prime.
		

Crossrefs

Cf. A000040, A380943, A383811, A383812, A383813, A383814, A383815, A383816.

Programs

Mathematica

f[n_] := Block[{cnt = 0, id = IntegerDigits@ n, k = 1, len, p, q, qp}, len = Length@ id; While[k < len, p = Take[id, k]; q = Take[id, -len + k]; qp = FromDigits[ Join[q, p]]; If[ PrimeQ[FromDigits[p]] && PrimeQ[FromDigits[q]] && PrimeQ[qp] && IntegerLength[qp] == len, cnt++]; k++]; cnt]; Select[ Prime@ Range@ 10000, f@# > 1 &]