A077794 Odd integers k such that 10^k - 1 - 10^((k-1)/2) is a prime of the form 9...989...9, called a palindromic wing prime or a near-repdigit palindromic prime.

53, 757, 2493, 3597, 5835, 46069, 95019, 104281, 134809

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.

The corresponding primes have a(n) digits all of which are '9's except for the middle digit which is an '8'. They are too large to be listed in a sequence on their own, cf. examples. See A077775-A077798 and A107123-A107127 for palindromic wing/near-repdigit primes with other digits. - M. F. Hasler, Mar 03 2019

1888529 is a term but its position is not known. - Jeppe Stig Nielsen, Jan 12 2024

a(10) > 600000. - Serge Batalov, Jan 17 2024

Examples

			a(1) = 53 corresponds to the 53-digit prime
  p = 99999999999999999999999999899999999999999999999999999.
a(2) = 757 corresponds to p = (10^757 - 1) - 10^378 = 99...99899...99.
		

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 99...99899...99
• PrimePages, Database Search Output.
• 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 - 1*10^Floor[n/2] - 1], Print[n]], {n, 3, 104300, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

PARI

is(n)=bittest(n,0)&&ispseudoprime(10^n-1-10^(n\2))
forstep(n=1,oo,2,is(n)&&print1(n",")) \\ M. F. Hasler, Mar 03 2019

Formula

a(n) = 2*A183187(n) + 1.

Extensions

a(9) from PWP table, added by Patrick De Geest, Nov 05 2014