A331862 Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

3, 26, 186, 206, 258, 3486, 12602

Offset: 1

Author

M. F. Hasler, Jan 30 2020

Keywords

• nonn
• base
• hard
• more

Comments

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

In base 10, R(n) - 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 0, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 0 is just left to the middle of the number.

There can't be an odd term > 3 because the corresponding palindrome factors as R((n-1)/2)*(10^((n+1)/2) + 1).

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

Examples

			For n = 3, R(n) - 10^floor(n/2) = 101 is prime.
For n = 26, R(n) - 10^floor(n/2) = 11111111111101111111111111 is prime.
		

Links

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

Crossrefs

Cf. A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A105992 (near-repunit primes), A138148 (Cyclops numbers with digits 0 & 1).

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

Programs

PARI

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

Extensions

a(6)-a(7) from Giovanni Resta, Jan 31 2020