A077798 -id:A077798 - cp's OEIS frontend

A077775 Odd numbers k such that (10^k - 1)/3 - 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime) of the form 3...313...3.

3, 7, 15, 123, 181, 185, 539, 597, 643, 743, 1553, 3135, 4769, 5133, 6177, 11733, 16103, 18997, 25271, 49025, 65043, 87965

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.
a(23) > 2*10^5. - Robert Price, Jan 29 2016
Primes of the form (10^k-1)/3 - 2*10^floor(k/2) are obtained for k in (2, 3, 6, 7, 8, 10, 15, 22, 34, 123, 126, 144, 181, 185, 198, 534, 539, 597, 606, ...). For example (10^2 - 1)/3 - 2*10^1 = 13 is also prime. However, for even k the result is not palindromic. See A077775-A077798, A107123-A107127 for PWP's with digits other than 3 and 1. - M. F. Hasler, Mar 03 2019

			a(3) = 15 corresponds to the prime (10^15 - 1)/3 - 2*10^7 = 333333313333333.

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 33...33133...33
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(10^n - 6*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 49100, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

is(n)=bittest(n,0)&&ispseudoprime(10^n\3-2*10^(n\2)) \\ M. F. Hasler, Mar 03 2019

a(n) = 2*A183174(n) + 1.

a(21)-a(22) from Robert Price, Jan 29 2016
a(21) corrected by Robert Price, Feb 03 2016
Name corrected by Jon E. Schoenfield, Oct 31 2018
Name edited by M. F. Hasler, Mar 03 2019

A183187 Numbers k such that 10^(2k+1)-10^k-1 is prime.

Original entry on oeis.org

26, 378, 1246, 1798, 2917, 23034, 47509, 52140, 67404

Offset: 1

Ray Chandler, Dec 28 2010

944264 is a term but its position is not known. - Jeppe Stig Nielsen, Jan 12 2024

a(10) > 300000. - Serge Batalov, Jan 17 2024

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...99899...99
PrimePages, Database Search Output.
Index entries for primes involving repunits.

Cf. A004023, A077775-A077798, especially A077794, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

Do[If[PrimeQ[10^(2n + 1) - 10^n - 1], Print[n]], {n, 3000}]

for(n=1,1e3,if(ispseudoprime(10^(2*n+1)-10^n-1),print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077794(n)-1)/2.

A115073 Numbers k such that 10^(2k+1)-710^k-1 is prime.

Original entry on oeis.org

1, 8, 9, 352, 530, 697, 1315, 1918, 2874, 5876, 6768, 62938, 134739

Offset: 1

N. J. A. Sloane, Mar 03 2006

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...99299...99
Index entries for primes involving repunits.

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

Do[If[PrimeQ[10^(2n + 1) - 7*10^n - 1], Print[n]], {n, 3000}]

for(n=0,1e4,if(ispseudoprime(t=10^(2*n+1)-7*10^n-1),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077778(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010

Added two more terms from PWP table, by Patrick De Geest, Nov 05 2014

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

A107648 Numbers n such that (10^(2n+1)+63*10^n-1)/9 is prime.

Original entry on oeis.org

1, 4, 6, 7, 384, 666, 675, 3165, 131020

Offset: 1

Farideh Firoozbakht, May 19 2005

n is in the sequence iff the palindromic number 1(n).8.1(n) is prime (dot between numbers means concatenation). Let f(n)=(10^(2n+1)+63*10^n-1)/9 then for all nonnegative integers m we have: I. 3 divides f(3m+2) II. 19 divides f(18m+13) III. 29 divides f(28*m+16) & 29 divides f(28*m+25) IV. 31 divides f(30*m+2) & 31 divides f(30*m+17) V. 41 divides f(5m+3), etc. So if n is in the sequence then n is not of the forms 3m+2, 18m+13, 28m+16 28m+25, 30m+2, 30m+17, 5m+3, etc.

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

			7 is in the sequence because (10^15+63*10^7-1)/9=1(7).8.1(7)=111111181111111 is prime.
666 is in the sequence because (10^(2*666+1)+63*10^666-1)/9=1(666).8.1(666) is prime.

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 141.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 11...11811...11
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 63*10^n - 1)/9], Print[n]], {n, 4000}]

for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+63*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077791(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010

a(9) from Robert Price, Aug 03 2024

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

Farideh Firoozbakht, May 19 2005

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

			26 is in the sequence because (10^(2*26+1)+72*10^26-1)/9=1(26).9.1(26)
= 11111111111111111111111111911111111111111111111111111 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...11911...11
Index entries for primes involving repunits.

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

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 *)

for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+72*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

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

Edited by Ray Chandler, Dec 28 2010

a(8)-a(9) from Robert Price, Sep 28 2017

A183174 Numbers k such that (10^(2k+1) - 6*10^k - 1)/3 is prime.

Original entry on oeis.org

1, 3, 7, 61, 90, 92, 269, 298, 321, 371, 776, 1567, 2384, 2566, 3088, 5866, 8051, 9498, 12635, 24512, 32521, 43982

Offset: 1

Ray Chandler, Dec 28 2010

a(23) > 10^5. - Robert Price, Jan 29 2016

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 33...33133...33.
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) - 6*10^n - 1)/3], Print[n]], {n, 3000}]

for(n=1,1e3,if(ispseudoprime((10^(2*n+1)-6*10^n-1)/3),print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077775(n) - 1)/2.

a(21)-a(22) from Robert Price, Jan 29 2016

A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.

Original entry on oeis.org

8, 181, 11811, 1118111, 111181111, 11111811111, 1111118111111, 111111181111111, 11111111811111111, 1111111118111111111, 111111111181111111111, 11111111111811111111111, 1111111111118111111111111, 111111111111181111111111111, 11111111111111811111111111111, 1111111111111118111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107648 = {1, 4, 6, 7, 384, 666, ...} 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)8(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11811...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077791-1)/2 = A107648: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes), A077798 (palindromic wing primes), A088281 (primes 1..1x1..1), A068160 (smallest of given length), A053701 (vertically symmetric numbers).
Cf. A332128 .. A332178, A181965 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332118 := n -> (10^(2*n+1)-1)/9+7*10^n;

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

apply( {A332118(n)=10^(n*2+1)\9+7*10^n}, [0..15])

def A332118(n): return 10**(n*2+1)//9+7*10**n

a(n) = A138148(n) + 8*10^n = A002275(2n+1) + 7*10^n.
G.f.: (8 - 707*x + 600*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

cp's OEIS Frontend

A077775 Odd numbers k such that (10^k - 1)/3 - 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime) of the form 3...313...3.

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A183187 Numbers k such that 10^(2k+1)-10^k-1 is prime.

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A115073 Numbers k such that 10^(2*k+1)-7*10^k-1 is prime.

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

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

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A107648 Numbers n such that (10^(2n+1)+63*10^n-1)/9 is prime.

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A115073 Numbers k such that 10^(2k+1)-710^k-1 is prime.

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