A331868 Numbers k for which R(k) + 4*10^floor(k/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

4, 147, 270, 1288, 1551, 3427

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) + 4*10^floor(n/2-1) has ceiling(n/2) digits 1, one digit 5, and again floor(n/2-1) digits 1. For odd and even n as well, the digit 5 appears just to the right of the middle of the number.

a(7) > 10^4. - Daniel Suteu, Feb 10 2020

a(7) > 5*10^4. - Michael S. Branicky, Nov 02 2024

Examples

			For n = 4, R(4) + 4*10^floor(4/2-1) = 1151 is prime.
For n = 5, R(5) + 4*10^floor(5/2-1) = 11151 =  3^3*7*59 is not prime.
For n = 147, R(147) + 4*10^72 = 1(74)51(72) is prime, where (.) indicates how many times the preceding digit is repeated.
		

Links

• Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
• Index to OEIS entries related to primes involving repdigits.

Crossrefs

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

Cf. A331863, A331860, A331864, A331867 (variants with digit 0, 2, 3 resp. 4 instead of 5), A331869 (variant with floor(n/2) instead of floor(n/2-1)).

Programs

Mathematica

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

PARI

for(n=2,9999,isprime(p=10^n\9+4*10^(n\2-1))&&print1(n","))

Extensions

a(6) from Daniel Suteu, Feb 10 2020