A331861 Numbers n for which R(n) + 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

1, 6, 10, 18, 24, 4978

Offset: 1

Author

M. F. Hasler, Jan 30 2020

Keywords

• nonn
• more
• hard
• base

Comments

The primes corresponding to the terms of the sequence are a subset of the near-repunit primes A105992.

In base 10, R(n) + 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 2, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 2 is just left to the middle of the number.

There cannot be an odd term > 1 since the corresponding palindrome factors as R((n+1)/2)*(10^((n-1)/2) + 1).

No term can be congruent to 2 mod 3. - Chai Wah Wu, Feb 07 2020

Examples

			For n = 1, R(n) + 10^floor(n/2) = 2 is prime.
For n = 6, R(n) + 10^floor(n/2) = 112111 is prime.
For n = 10, R(n) + 10^floor(n/2) = 1111211111 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. A331860 (variant with floor(n/2-1) instead of floor(n/2)), A331862 (variant with - (digit 0) instead of + (digit 2)).

Programs

PARI

for(n=0,9999,isprime(p=10^n\9+10^(n\2))&&print1(n","))

Extensions

a(6) from Daniel Suteu, Feb 01 2020