A383812 Primes which satisfy the requirements of A380943 in exactly three ways.

19937, 103997, 377477, 577937, 738677, 739397, 877937, 2116397, 3110273, 3314513, 3343337, 3634313, 3833359, 5935393, 7147397, 7276337, 7511033, 7699157, 7723337, 11816911, 14713613, 19132213, 19132693, 19998779, 22739317, 23201359, 31189757, 31614377, 31669931, 31687151

Offset: 1

Author

James C. McMahon and Robert G. Wilson v, May 18 2025

Keywords

• nonn
• base

Comments

The requirements of A380943 are that primes, p_n, 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, 0, 0, 1, 7, 19, 70, 299, 1872, ..., .

Crossrefs

Cf. A000040, A238499, A380943, A383810, A383811, 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@ 1980000, f@# == 3 &]