A331865 Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

0, 1, 2, 3, 5, 6, 8, 12, 20, 39, 74, 78, 80, 84, 104, 195, 654, 980, 2076, 5940, 19637

Offset: 1

Author

M. F. Hasler, Jan 30 2020

Keywords

• nonn
• base
• hard
• more

Comments

The corresponding primes are a subsequence of A105992: near-repunit primes.

In base 10, R(n) + 2*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 3, and again floor(n/2) digits 1 (except for n=0). For odd n, this is a palindrome (a.k.a. wing prime, cf. A077779), for even n the digit 3 is just left to the middle of the number.

a(22) > 50000. - Michael S. Branicky, Feb 19 2025

Examples

			For n = 0, R(0) + 2*10^floor(0/2) = 2 is prime.
For n = 1, R(1) + 2*10^floor(1/2) = 3 is prime.
For n = 2, R(2) + 2*10^floor(2/2) = 31 is prime.
For n = 3, R(3) + 2*10^floor(3/2) = 131 is prime.
For n = 5, R(5) + 2*10^floor(5/2) = 11311 is prime.
For n = 6, R(6) + 2*10^floor(6/2) = 113111 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. A331860 & A331863 (variants with digit 2 resp. 0 instead of 3), A331864 (variant with floor(n/2-1) instead of floor(n/2)).

Cf. A077779 (odd terms).

Programs

Mathematica

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 2*10^Floor[#/2]] &] (* Michael De Vlieger, Jan 31 2020 *)

PARI

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

Extensions

a(18)-a(20) from Giovanni Resta, Jan 30 2020

a(21) from Michael S. Branicky, Feb 19 2025