A331869 Numbers n for which R(n) + 4*10^floor(n/2) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

1, 3, 4, 15, 76, 91, 231, 1363, 1714, 1942, 2497, 4963, 5379, 12397, 23224, 26395

Offset: 1

Author

M. F. Hasler, Feb 09 2020

Keywords

• nonn
• base
• hard
• more

Comments

For n > 1, the corresponding primes are a subset of A105992: near-repunit primes.

In base 10, R(n) + 4*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 5, and again floor(n/2) digits 1, except for n = 0. For odd n, this is a palindrome (a.k.a. wing prime, cf. A077783: subsequence of odd terms), for even n the digit 5 is just left to the middle of the number.

See also the variant A331868 where the digit 5 is just to the right of the middle.

Examples

			For n = 1, R(1) + 4*10^floor(1/2) = 5 is prime.
For n = 3, R(3) + 4*10^floor(3/2) = 151 is prime.
For n = 4, R(4) + 4*10^floor(4/2) = 1511 is prime.
For n = 15, R(15) + 4*10^floor(15/2) = 111111151111111 is prime.
		

Links

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

Crossrefs

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).

Cf. A331862, A331861, A331865, A331866 (variants with digit 0, 2, 3 or 4 instead of 5), A331868 (variant with floor(n/2-1) instead of floor(n/2)).

Cf. A077783 (odd terms).

Programs

Mathematica

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 4*10^Floor[#/2]] &]

PARI

for(n=0,9999,ispseudoprime(p=10^n\9+4*10^(n\2))&&print1(n","))

Extensions

a(12)-a(14) from Michael S. Branicky, Feb 03 2023

a(15)-a(16) from Michael S. Branicky, Apr 11 2023