A383815 Palindromic primes in A380943.

313, 373, 797, 11311, 13331, 13931, 17971, 19991, 31013, 35353, 36263, 36563, 38783, 71317, 79397, 97379, 98389, 1129211, 1196911, 1611161, 1793971, 1982891, 3106013, 3166613, 3193913, 3236323, 3288823, 3304033, 3319133, 3329233, 3365633, 3417143, 3447443, 3449443, 3515153, 3670763

Offset: 1

Author

James C. McMahon and Robert G. Wilson v, Jun 06 2025

Keywords

• base
• nonn

Comments

A380943 requires 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.

Intersection of A002385 and A380943.

Examples

			The palindromic prime 313 is formed by the concatenation of the primes 31 and 3, which reversed, also form the prime 331. The palindromic prime 13931 is formed by the concatenation of 139 and 31; 31139 is also prime.
		

Crossrefs

Cf. A000040, A002385, A105184, A380943, A383810, A383811, A383812, A383813.

Programs

Maple

rev:= proc(n) local L,i;
   L:= convert(n,base,10);
   add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(x,y) y + 10^(ilog10(y)+1)*x end proc:
filter:= proc(z) local i,x,y;
  if not isprime(z) then return false fi;
  for i from 1 to ilog10(z) do
    x:= z mod 10^i;
    if x < 10^(i-1) then next fi;
    y:= (z-x)/10^i;
    if isprime(x) and isprime(y) and isprime(tcat(x,y)) then return true fi;
  od;
  false
end proc:
N:= 7: # for terms of up to 7 digits
R:= NULL:
for d from 1 to (N-1)/2 do
  for x from 10^(d-1) to 10^d-1 do
    for y from 0 to 9 do
      z:= rev(x) + 10^d * y + 10^(d+1)*x;
      if filter(z) then R:= R,z fi
od od od:
R;  # Robert Israel, Jun 08 2025

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]; fQ[n_] := Reverse[idn = IntegerDigits@ n] == idn && f@ n > 0; Select[ Prime@ Range@ 264000, fQ]