A107124 Numbers k such that (10^(2*k+1)+27*10^k-1)/9 is prime.

2, 3, 32, 45, 1544, 149610

Offset

1,1

Comments

k is in the sequence iff the palindromic number 1(k).4.1(k) is prime (dot between numbers means concatenation). If k is in the sequence then k is not of the forms 3m+1, 16m+11, 16m+12, 18m+11, 18m+15, etc. (the proof is easy).

References

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

Links

Table of n, a(n) for n=1..6.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)

Makoto Kamada, Prime numbers of the form 11...11411...11

Index entries for primes involving repunits.

Formula

a(n) = (A077780(n)-1)/2.

Example

32 is in the sequence because the palindromic number (10^(2*32+1)+27*10^32-1)/9 = 1(32).4.1(32) =
11111111111111111111111111111111411111111111111111111111111111111 is prime.

Mathematica

Do[If[PrimeQ[(10^(2n + 1) + 27*10^n - 1)/9], Print[n]], {n, 2200}]
Select[Range[1600], PrimeQ[FromDigits[Join[PadRight[{}, #, 1], {4}, PadRight[ {}, #, 1]]]]&] (* Harvey P. Dale, Aug 01 2017 *)

Prog

(PARI) is(n)=ispseudoprime((10^(2*n+1)+27*10^n-1)/9) \\ Charles R Greathouse IV, May 22 2017

Crossrefs

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

Sequence in context: A041053 A212763 A103108 * A141049 A354612 A052830

Adjacent sequences: A107121 A107122 A107123 * A107125 A107126 A107127

Keyword

nonn,base,more

Author

Farideh Firoozbakht, May 19 2005

Extensions

Edited by Ray Chandler, Dec 28 2010

a(6) from Robert Price, Jun 12 2026

Status

approved