A077789 -id:A077789 - cp's OEIS frontend

A107127 Numbers n such that (10^(2n+1)+54*10^n-1)/9 is prime.

0, 3, 33, 311, 2933, 22235, 39165, 41585

Offset: 1

Farideh Firoozbakht, May 19 2005

n is in the sequence iff the palindromic number 1(n).7.1(n) is prime (dot between numbers means concatenation). If n is in the sequence then n is not of the forms 3m+1, 6m, 6m+2, 7m+2, 16m+9, 16m+14, 18m+1, 18m+7, 22m+13, 22m+19, etc. (the proof is easy).

a(9) > 10^5. - Robert Price, Apr 30 2017

			3 is in the sequence because (10^(2*3+1)+54*10^3-1)/9=1(3).7.1(3)=1117111 is prime.
2933 is in the sequence because (10^(2*2933+1)+54*10^2933-1)/9=1(2933).7.1(2933) 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...11711...11
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 54*10^n - 1)/9], Print[n]], {n, 3250}]

for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+54*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077789(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010

a(6)-a(8) from Robert Price, Apr 30 2017

A332117 a(n) = (10^(2n+1)-1)/9 + 6*10^n.

Original entry on oeis.org

7, 171, 11711, 1117111, 111171111, 11111711111, 1111117111111, 111111171111111, 11111111711111111, 1111111117111111111, 111111111171111111111, 11111111111711111111111, 1111111111117111111111111, 111111111111171111111111111, 11111111111111711111111111111, 1111111111111117111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107127 = {0, 3, 33, 311, 2933, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (1)7(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11711...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077789-1)/2 = A107127: 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. A332127 .. A332197 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332117 := n -> (10^(2*n+1)-1)/9+6*10^n;

Array[(10^(2 # + 1)-1)/9 + 6*10^# &, 15, 0]

apply( {A332117(n)=10^(n*2+1)\9+6*10^n}, [0..15])

def A332117(n): return 10**(n*2+1)//9+6*10**n

a(n) = A138148(n) + 7*10^n = A002275(2n+1) + 6*10^n.
G.f.: (7 - 606*x + 500*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.

cp's OEIS Frontend

A107127 Numbers n such that (10^(2n+1)+54*10^n-1)/9 is prime.

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