A107123 Numbers k such that (10^(2*k+1)+18*10^k-1)/9 is prime.

0, 1, 2, 19, 97, 9818

Offset: 1

Author

Farideh Firoozbakht, May 19 2005

Keywords

• nonn
• base
• more

Comments

A number k is in the sequence iff the palindromic number 1(k).3.1(k) is prime (1(k) means k copies of 1; dot between numbers means concatenation). If k is a positive term of the sequence then k is not of the form 3m, 6m+4, 12m+10, 28m+5, 28m+8, etc. (the proof is easy).

The palindromic number 1(k).2.1(k) is never prime for k > 0 because it is (1.0(k-1).1)*(1(k+1)). - Robert Israel, Jun 11 2015

a(7) > 10^5. - Robert Price, Apr 02 2016

Examples

			19 is in the sequence because the palindromic number (10^(2*19+1)+18*10^19-1)/9 = 1(19).3.1(19) = 111111111111111111131111111111111111111 is prime.
		

References

• C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Links

• Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
• Makoto Kamada, Prime numbers of the form 11...11311...11
• Index entries for primes involving repunits.

Crossrefs

Cf. A004023, A077775-A077798, A107124, A107125, A107126, A107127, A107648, A107649, A115073, A183174-A183187.

Programs

Maple

select(n -> isprime((10^(2*n+1)+18*10^n-1)/9), [$0..100]); # Robert Israel, Jun 11 2015

Mathematica

Do[If[PrimeQ[(10^(2n + 1) + 18*10^n - 1)/9], Print[n]], {n, 2500}]

PARI

for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+18*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

Formula

a(n) = (A077779(n-1)-1)/2, for n > 1. [Corrected by M. F. Hasler, Feb 06 2020]

Extensions

Edited by Ray Chandler, Dec 28 2010