A183184 -id:A183184 - cp's OEIS frontend

A332191 a(n) = 10^(2n+1) - 1 - 8*10^n.

1, 919, 99199, 9991999, 999919999, 99999199999, 9999991999999, 999999919999999, 99999999199999999, 9999999991999999999, 999999999919999999999, 99999999999199999999999, 9999999999991999999999999, 999999999999919999999999999, 99999999999999199999999999999, 9999999999999991999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

See A183184 = {1, 5, 13, 43, 169, 181, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)1(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 9999199...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077776-1)/2 = A183184: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332121 .. A332181 (variants with different repeated digit 2, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Maple
```
A332191 := n -> 10^(n*2+1)-1-8*10^n;
```
Mathematica
```
Array[ 10^(2 # + 1)-1-8*10^# &, 15, 0]
```

apply( {A332191(n)=10^(n*2+1)-1-8*10^n}, [0..15])

def A332191(n): return 10**(n*2+1)-1-8*10^n

a(n) = 9*A138148(n) + 10^n.
G.f.: (1 + 808*x - 1700*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.

A077776 Numbers k such that (10^k - 1) - 8*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 11, 27, 87, 339, 363, 3159, 36155, 45305, 314727

Offset: 1

Patrick De Geest, Nov 16 2002

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

			27 is a term because (10^27 - 1) - 8*10^13 = 999999999999919999999999999.

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 99...99199...99
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[10^n - 8*10^Floor[n/2] - 1], Print[n]], {n, 3, 1000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183184(n) + 1.

One more term from PWP table added by Patrick De Geest, Nov 05 2014

Name corrected by Jon E. Schoenfield, Oct 31 2018

A332181 a(n) = 8(10^(2n+1)-1)/9 - 710^n.

Original entry on oeis.org

1, 818, 88188, 8881888, 888818888, 88888188888, 8888881888888, 888888818888888, 88888888188888888, 8888888881888888888, 888888888818888888888, 88888888888188888888888, 8888888888881888888888888, 888888888888818888888888888, 88888888888888188888888888888, 8888888888888881888888888888888

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077776-1)/2 = A183184: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

A332181 := n -> 8*(10^(2*n+1)-1)/9-7*10^n;

Array[8 (10^(2 # + 1)-1)/9 - 7*10^# &, 15, 0]

apply( {A332181(n)=10^(n*2+1)\9*8-7*10^n}, [0..15])

def A332181(n): return 10**(n*2+1)//9*8-7*10**n

a(n) = 8*A138148(n) + 10^n = A002282(2n+1) - 7*10^n.
G.f.: (1 + 707*x - 1500*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.

A331815 Numbers k such that 10^(2k) - 810^(k-1) - 1 is prime.

Original entry on oeis.org

3, 4, 132, 471, 1935, 4258, 9444

Offset: 1

Eder Vanzei, Jan 27 2020

Also numbers k such that the concatenation (k 9's)1(k-1 9's) is prime.

			3 is a term because 999199 is prime.
4 is a term because 99991999 is prime.

Index to OEIS entries related to primes involving repdigits.

Cf. A000040.

Cf. A077776 = A183184*2+1: palindromic near-repdigit primes 9..919..9.

Select[Range[500], PrimeQ[10^(2*#) - 8*10^(#-1) - 1] &] (* Amiram Eldar, Jan 28 2020 *)

(is_A331815(n)=ispseudoprime(100^n-8*10^(n-1)-1)); for(n=1, 9999, is_A331815(n)&&print1(n", "))

a(7) from Giovanni Resta, Jan 28 2020

cp's OEIS Frontend

A332191 a(n) = 10^(2n+1) - 1 - 8*10^n.

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A077776 Numbers k such that (10^k - 1) - 8*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A332181 a(n) = 8*(10^(2n+1)-1)/9 - 7*10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A331815 Numbers k such that 10^(2*k) - 8*10^(k-1) - 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A332181 a(n) = 8(10^(2n+1)-1)/9 - 710^n.

A331815 Numbers k such that 10^(2k) - 810^(k-1) - 1 is prime.