1,1

"Reversible prime" means a palindromic (here: wing) prime (A077798 c A002385) or an emirp (cf. A006567). These primes are also near-repdigit primes here, with all k digits 9 except for one digit 4.

The definition implies that 0 < m < k, and WLOG we can assume m < k/2; then 10^k - 5*10^m - 1 is the larger and 10^k - 5*10^(k-m-1) - 1 is the smaller of the two emirps, unless 2m = k-1, in which case its a palindromic wing prime (PWP).

For k = 15, m = 5, we have 10^15 - 1 - 5*10^8 = 999999999599999, an emirp, with reverse prime 999995999999999.

(PARI) is_A393132(k, r=10^k-1)=for(m=1, k\/2-1, ispseudoprime(r-10^(k-m-1)*5)&& ispseudoprime(r-10^m*5)&& return(m)) /* return the m-value if it exists, else 0 */

for(k=1, oo, is_A393132(k) && print1(k", "))

Cf. A393126, A393129, A393130 (similar with distinct digit 8, 7 and 5, respectively).

Cf. A006567 (emirps), A004086 (n with digits reversed), A173594 (near-repdigit emirps).

Cf. A002385 (palindromic primes), A077798 (palindromic wing primes).

Sequence in context: A290749 A374005 A091017 * A157716 A113968 A093812

Adjacent sequences: A393129 A393130 A393131 * A393133 A393134 A393135

nonn,base,hard,more,new

M. F. Hasler, Mar 24 2026

a(10)-a(12) from Michael S. Branicky, Mar 28 2026

approved