cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A107649 Numbers n such that (10^(2n+1)+72*10^n-1)/9 is prime.

Original entry on oeis.org

1, 4, 26, 187, 226, 874, 13309, 34016, 42589
Offset: 1

Views

Author

Farideh Firoozbakht, May 19 2005

Keywords

Comments

n is in the sequence iff the palindromic number 1(n).9.1(n) is prime (dot between numbers means concatenation). If n is in the sequence then n is not of the forms 3m, 6m+5, 22m+3, 22m+7, etc. (the proof is easy).
a(10) > 200000. - _Robert Price, Jan 30 2025

Examples

			26 is in the sequence because (10^(2*26+1)+72*10^26-1)/9=1(26).9.1(26)
= 11111111111111111111111111911111111111111111111111111 is prime.
		

References

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

Crossrefs

Programs

  • Mathematica
    Do[If[PrimeQ[(10^(2n + 1) + 72*10^n - 1)/9], Print[n]], {n, 3000}]
    prQ[n_]:=Module[{c=PadRight[{},n,1]},PrimeQ[FromDigits[Join[c,{9},c]]]]; Select[Range[13500],prQ] (* Harvey P. Dale, Jan 19 2014 *)
  • PARI
    for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+72*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

Formula

a(n) = (A077795(n)-1)/2.

Extensions

Edited by Ray Chandler, Dec 28 2010
a(8)-a(9) from Robert Price, Sep 28 2017

A332119 a(n) = (10^(2n+1)-1)/9 + 8*10^n.

Original entry on oeis.org

9, 191, 11911, 1119111, 111191111, 11111911111, 1111119111111, 111111191111111, 11111111911111111, 1111111119111111111, 111111111191111111111, 11111111111911111111111, 1111111111119111111111111, 111111111111191111111111111, 11111111111111911111111111111, 1111111111111119111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107649 = {1, 4, 26, 187, 226, 874, ...} for the indices of primes.

Crossrefs

Cf. (A077795-1)/2 = A107649: 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. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332112 .. A332118 (variants with different middle digit 2, ..., 8).

Programs

  • Maple
    A332119 := n -> (10^(2*n+1)-1)/9+8*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 8*10^# &, 15, 0]
    Table[FromDigits[Join[PadRight[{},n,1],{9},PadRight[{},n,1]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,191,11911},20] (* Harvey P. Dale, Mar 30 2024 *)
  • PARI
    apply( {A332119(n)=10^(n*2+1)\9+8*10^n}, [0..15])
    
  • Python
    def A332119(n): return 10**(n*2+1)//9+8*10**n

Formula

a(n) = A138148(n) + 9*10^n = A002275(2n+1) + 8*10^n.
G.f.: (9 - 808*x + 700*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.
Showing 1-2 of 2 results.