A331864 Numbers k such that R(k) + 2*10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

2, 3, 5, 8, 9, 39, 78, 81, 155, 249, 387, 395, 510, 711, 1173, 1751, 10245

Offset: 1

Author

M. F. Hasler, Jan 30 2020

Keywords

• nonn
• base
• hard
• more

Comments

The corresponding primes are near-repunit primes, cf. A105992.

In base 10, R(k) + 2*10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 3 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 3 just left of the middle of the repunit of length k.

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

Examples

			For k = 2, R(2) + 2*10^(1-1) = 13 is prime.
For k = 3, R(3) + 2*10^(1-1) = 113 is prime.
For k = 5, R(5) + 2*10^(2-1) = 11131 is prime.
For k = 8, R(8) + 2*10^(4-1) = 11113111 is prime.
		

Links

• Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).

Cf. A331865 (variant with floor(n/2) instead of floor(n/2-1)), A331860, A331863 (variants with digit 2 resp. 0 instead of digit 3).

Programs

PARI

for(n=2,999,isprime(p=10^n\9+2*10^(n\2-1))&&print1(n","))

Extensions

a(13)-a(16) from Daniel Suteu, Feb 01 2020

a(17) from Michael S. Branicky, Feb 03 2023