A129613 -id:A129613 - cp's OEIS frontend

A233464 a(n) is the smallest natural number m such that 10^10^k + m is prime for k = 0, 1, ...., n.

1, 19, 5641, 1289743, 2578966671

Offset: 0

Farideh Firoozbakht, Mar 13 2014

			29 (=10^1+19) and 10000000019 (=10^10+19) are primes so a(1)=19.

Cf. A129613, A129614, A129615.

okm(m, n) = {for (k=0, n, if (!isprime(10^10^k + m), return (0)););return (1);}
a(n) = {m = 0; while (!okm(m, n), m++); m;} \\ Michel Marcus, Mar 16 2014

a(4) from Giovanni Resta, Mar 14 2014

A371303 Numbers k > 4 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every natural m > 0 with 2^(2^m) < k.

Original entry on oeis.org

7, 9, 15, 27, 57, 63, 195, 267, 363, 405, 483, 603, 1197, 1233, 1443, 1737, 2715, 4257, 5403, 6117, 21855, 22287, 26817, 40755, 63777, 260007, 617253, 986733, 1151655, 1167837, 1174503, 1199373, 1331595, 3233307, 4128873, 4138707, 4609527, 5938107, 7203945, 7605213, 8379405, 8587545, 9596223

Offset: 1

Thomas Ordowski, Mar 18 2024

nonn

It seems that there are infinitely many such numbers.
If k > 7 is such a number, then it is odd and divisible by 3.
Conjecture: numbers k > 2 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every integer m >= 0 with 2^(2^m) < k are only 9, 15, and 195 (Amiram Eldar checked that there are no more terms k < 10^8).

Cf. A039669, A129613, A370523.

q[k_] := Module[{m = 1}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)] && PrimeQ[k + 2^(2^m)], m++]; 2^(2^m) > k]; Select[Range[5, 10^6, 2], q] (* Amiram Eldar, Mar 18 2024 *)

More terms from Amiram Eldar, Mar 18 2024

cp's OEIS Frontend

A233464 a(n) is the smallest natural number m such that 10^10^k + m is prime for k = 0, 1, ...., n.

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