A331863 Numbers k such that R(k) - 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

8, 12, 17, 20, 24, 42, 1124, 1169, 1538, 7902, 27617, 29684

Offset: 1

Author

M. F. Hasler, Jan 30 2020

Keywords

• nonn
• hard
• more
• base

Comments

The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.

In base 10, R(k) - 10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 0 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 0 just right of the middle of the repunit of length k.

No term can be congruent to 1 (mod 3). - Chai Wah Wu, Feb 07 2020

a(13) > 50000. - Michael S. Branicky, Jul 23 2024

Examples

			For k = 8,  R(8)  - 10^(4-1) = 11110111 is prime.
For k = 12, R(12) - 10^(6-1) = 111111011111 is prime.
For k = 17, R(12) - 10^(8-1) = 11111111101111111 is prime.
		

Crossrefs

Cf. A002275 (repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A138148 (Cyclop numbers with digits 0 & 1).

Cf. A331862 (variant with floor(n/2) instead of floor(n/2-1)), A331860 (variant with + (digit 2) instead of - (digit 0)).

Programs

PARI

for(n=2,9999,isprime(p=10^n\9-10^(n\2-1))&&print1(n","))

Extensions

a(7)-a(10) from Giovanni Resta, Jan 31 2020

a(11)-a(12) from Michael S. Branicky, Jul 22 2024