A107126 -id:A107126 - cp's OEIS frontend

A107123 Numbers k such that (10^(2k+1)+1810^k-1)/9 is prime.

0, 1, 2, 19, 97, 9818

Offset: 1

Farideh Firoozbakht, May 19 2005

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

			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.

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

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.

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

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

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

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

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

Edited by Ray Chandler, Dec 28 2010

A332116 a(n) = (10^(2n+1)-1)/9 + 5*10^n.

Original entry on oeis.org

6, 161, 11611, 1116111, 111161111, 11111611111, 1111116111111, 111111161111111, 11111111611111111, 1111111116111111111, 111111111161111111111, 11111111111611111111111, 1111111111116111111111111, 111111111111161111111111111, 11111111111111611111111111111, 1111111111111116111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107126 = {10, 14, 40, 59, 160, 412, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (1)6(1), updated: June 25, 2017.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Makoto Kamada, Factorization of 11...11611...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077706-1)/2 = A107126: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332126 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332116 := n -> (10^(2*n+1)-1)/9+5*10^n;

Array[(10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]

apply( {A332116(n)=10^(n*2+1)\9+5*10^n}, [0..15])

def A332116(n): return 10**(n*2+1)//9+5*10**n

a(n) = A138148(n) + 6*10^n = A002275(2n+1) + 5*10^n.
G.f.: (6 - 505*x + 400*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: exp(x)*(10*exp(99*x) + 45*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A077787 Numbers k such that (10^k - 1)/9 + 5*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

21, 29, 81, 119, 321, 825, 1121, 2579, 3693

Offset: 1

Patrick De Geest, Nov 16 2002

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

a(10) > 4*10^5. - _Robert Price, Jan 23 2025

			21 is a term because (10^21 - 1)/9 + 5*10^10 = 111111111161111111111.

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

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 11...11611...11
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(10^n + 45*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 4000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A107126(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A332116 a(n) = (10^(2n+1)-1)/9 + 5*10^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A077787 Numbers k such that (10^k - 1)/9 + 5*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A107123 Numbers k such that (10^(2k+1)+1810^k-1)/9 is prime.