A077775 Odd numbers k such that (10^k - 1)/3 - 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime) of the form 3...313...3.

3, 7, 15, 123, 181, 185, 539, 597, 643, 743, 1553, 3135, 4769, 5133, 6177, 11733, 16103, 18997, 25271, 49025, 65043, 87965

Offset: 1

Author

Patrick De Geest, Nov 16 2002

Keywords

• more
• nonn
• base

Comments

Prime versus probable prime status and proofs are given in the author's table.

a(23) > 2*10^5. - Robert Price, Jan 29 2016

Primes of the form (10^k-1)/3 - 2*10^floor(k/2) are obtained for k in (2, 3, 6, 7, 8, 10, 15, 22, 34, 123, 126, 144, 181, 185, 198, 534, 539, 597, 606, ...). For example (10^2 - 1)/3 - 2*10^1 = 13 is also prime. However, for even k the result is not palindromic. See A077775-A077798, A107123-A107127 for PWP's with digits other than 3 and 1. - M. F. Hasler, Mar 03 2019

Examples

			a(3) = 15 corresponds to the prime (10^15 - 1)/3 - 2*10^7 = 333333313333333.
		

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 33...33133...33
• Index entries for primes involving repunits.

Crossrefs

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

Programs

Mathematica

Do[ If[ PrimeQ[(10^n - 6*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 49100, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

PARI

is(n)=bittest(n,0)&&ispseudoprime(10^n\3-2*10^(n\2)) \\ M. F. Hasler, Mar 03 2019

Formula

a(n) = 2*A183174(n) + 1.

Extensions

a(21)-a(22) from Robert Price, Jan 29 2016

a(21) corrected by Robert Price, Feb 03 2016

Name corrected by Jon E. Schoenfield, Oct 31 2018

Name edited by M. F. Hasler, Mar 03 2019