A183174 -id:A183174 - cp's OEIS frontend

A332131 a(n) = (10^(2n+1)-1)/3 - 2*10^n.

1, 313, 33133, 3331333, 333313333, 33333133333, 3333331333333, 333333313333333, 33333333133333333, 3333333331333333333, 333333333313333333333, 33333333333133333333333, 3333333333331333333333333, 333333333333313333333333333, 33333333333333133333333333333, 3333333333333331333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183174 = {1, 3, 7, 61, 90, 92, 269, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)1(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33133...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077775-1)/2 = A183174: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332121 .. A332191 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332131 := n -> (10^(2*n+1)-1)/3-2*10^n;

Array[3 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]

apply( {A332131(n)=10^(n*2+1)\3-2*10^n}, [0..15])

def A332131(n): return 10**(n*2+1)//3-2*10**n

a(n) = 3*A138148(n) + 1*10^n = A002277(2n+1) - 2*10^n.
G.f.: (1 + 202*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.

A077779 Numbers k such that (10^k - 1)/9 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 5, 39, 195, 19637

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.
a(6) > 2*10^5. - Robert Price, Apr 02 2016
The number k = 1 would also correspond to a prime, 3, but not "near-repdigit" or "wing" in a strict sense. - M. F. Hasler, Feb 09 2020

			5 is a term because (10^5 - 1)/9 + 2*10^2 = 11311.

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, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

See A332113 for the (prime and composite) near-repunit palindromes 1..131..1.

Do[ If[ PrimeQ[(10^n + 18*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 20000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A107123(n+1) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

A077780 Numbers k such that (10^k - 1)/9 + 3*10^floor(k/2) is palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

5, 7, 65, 91, 3089

Offset: 1

Patrick De Geest, Nov 16 2002

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

			7 is a term because (10^7 - 1)/9 + 3*10^3 = 1114111.

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...11411...11
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(10^n + 27*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 3100, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A107124(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

A077783 Numbers k such that (10^k-1)/9 + 4*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 15, 91, 231, 1363, 2497, 4963, 5379, 12397, 26395, 120253, 200145

Offset: 1

Patrick De Geest, Nov 16 2002

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

a(13) > 200466. - _Robert Price, Sep 05 2023

			15 is a term because (10^15 - 1)/9 + 4*10^7 = 111111151111111.

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...11511...11
Index entries for primes involving repunits.

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

[n: n in [3..2000 by 2] | IsPrime((10^n+36*10^(n div 2)-1) div 9)]; // Vincenzo Librandi, Oct 13 2015

Do[ If[ PrimeQ[(10^n + 36*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 26400, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A107125(n) + 1.

a(11) from Robert Price, Oct 12 2015
Name edited by Jon E. Schoenfield, Oct 13 2015
a(12) from Robert Price, Sep 05 2023

A107125 Numbers k such that (10^(2k+1) + 3610^k - 1)/9 is prime.

Original entry on oeis.org

0, 1, 7, 45, 115, 681, 1248, 2481, 2689, 6198, 13197, 60126, 100072

Offset: 1

Farideh Firoozbakht, May 19 2005

k is in the sequence iff the palindromic number 1(k).5.1(k) is prime (dot between numbers means concatenation). If k is in the sequence then k is not of the forms 3m+2, 18m+12, 18m+14, 22m+4, 22m+6, etc. (the proof is easy).

a(14) > 100233. - _Robert Price, Sep 05 2023

			1248 is in the sequence because (10^(2*1248+1)+36*10^1248-1)/9=1(1248).5.1(1248) 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...11511...11
Index entries for primes involving repunits.

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

[n: n in [0..700] | IsPrime((10^(2*n+1)+36*10^n-1) div 9)]; // Vincenzo Librandi, Oct 13 2015

Do[If[PrimeQ[(10^(2n + 1) + 36*10^n - 1)/9], Print[n]], {n, 2200}]

is(n)=ispseudoprime((10^(2*n+1)+36*10^n-1)/9) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = (A077783(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010
a(12) from Robert Price, Oct 12 2015
a(13) from Robert Price, Sep 05 2023

A107126 Numbers n such that (10^(2n+1)+45*10^n-1)/9 is prime.

Original entry on oeis.org

10, 14, 40, 59, 160, 412, 560, 1289, 1846

Offset: 1

Farideh Firoozbakht, May 19 2005

n is in the sequence iff the palindromic number 1(n).6.1(n) is prime (dot between numbers means concatenation). If n is in the sequence then n is not of the forms 3m, 6m + 1, 16m + 2, 16m + 5, 22m + 1, 22m + 9, etc. (the proof is easy).

a(10) > 200000 - Robert Price, Jan 23 2025

			14 is in the sequence because (10^(2*14+1)+45*10^14-1)/9=1(14).6.1(14) = 11111111111111611111111111111 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...11611...11
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 45*10^n - 1)/9], Print[n]], {n, 2500}]
Position[Table[FromDigits[Join[PadRight[{},n,1],{6},PadRight[{},n,1]]],{n,1850}],?PrimeQ]//Flatten (* _Harvey P. Dale, Jun 22 2017 *)

is(n)=ispseudoprime((10^(2*n+1)+45*10^n-1)/9) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = (A077787(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010.

A077777 Numbers k such that 7(10^k - 1)/9 - 510^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 7, 15, 21, 25, 961, 1899, 3891, 15097, 17847

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.
a(11) > 2*10^5. - Robert Price, Nov 02 2015
A183178(1) = 0 would correspond to an initial term 1 in this sequence which yields the prime 2 (which has a "wing" of length 0 and is a palindrome and repdigit but not near-repdigit). - M. F. Hasler, Feb 08 2020

			15 is a term because 7*(10^15 - 1)/9 - 5*10^7 = 777777727777777.

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 77...77277...77
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(7*10^n - 45*10^Floor[n/2] - 7)/9], Print[n]], {n, 3, 1000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

for(k=1,oo,ispseudoprime(10^k\9*7-5*10^(k\2))&&print1(k",")) \\ M. F. Hasler, Feb 08 2020

a(n) = 2*A183178(n+1) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

A077778 Numbers k such that (10^k - 1) - 7*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 17, 19, 705, 1061, 1395, 2631, 3837, 5749, 11753, 13537, 125877, 269479

Offset: 1

Patrick De Geest, Nov 16 2002

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

			17 is a term because (10^17 - 1) - 7*10^8 = 99999999299999999.

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^n - 7*10^Floor[n/2] - 1], Print[n]], {n, 3, 14600, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A115073(n) + 1.

Two more terms from PWP table added by Patrick De Geest, Nov 05 2014

Name corrected by Jon E. Schoenfield, Oct 31 2018

A077781 Numbers k such that 7(10^k - 1)/9 - 310^floor(k/2) is a palindromic wing prime (also known as near-repdigit palindromic prime).

Original entry on oeis.org

5, 7, 13, 47, 73, 139, 1123, 1447, 6877, 8209, 18041, 27955, 39311, 64801

Offset: 1

Patrick De Geest, Nov 16 2002

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

a(15) > 2*10^5. - Robert Price, Nov 23 2015

			7 is a term because 7*(10^7 - 1)/9 - 3*10^3 = 7774777.

C. Caldwell and H. Dubner, The near repdigit primes A(n-k-1)B(1)A(k), especially 9(n-k-1)8(1)9(k), 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 77...77477...77
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(7*10^n - 27*10^Floor[n/2] - 7)/9], Print[n]], {n, 3, 39400, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

for(n=1, 1e3, if(ispseudoprime((7*10^(2*n+1)-27*10^n-7)/9), print1(2*n+1, ", "))) \\ Altug Alkan, Nov 23 2015

a(n) = 2*A183179(n) + 1.

a(14) from Robert Price, Nov 23 2015

Name corrected by Jon E. Schoenfield, Oct 31 2018

A077782 Numbers k such that (10^k - 1) - 5*10^floor(m/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

29, 45, 73, 209, 2273, 35729, 50897

Offset: 1

Patrick De Geest, Nov 16 2002

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

			29 is a term because (10^29 - 1) - 5*10^14 = 99999999999999499999999999999.

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...99499...99
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[10^n - 5*10^Floor[n/2] - 1], Print[n]], {n, 3, 50900, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183185(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

cp's OEIS Frontend

A332131 a(n) = (10^(2n+1)-1)/3 - 2*10^n.

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A077779 Numbers k such that (10^k - 1)/9 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A077780 Numbers k such that (10^k - 1)/9 + 3*10^floor(k/2) is palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A077783 Numbers k such that (10^k-1)/9 + 4*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A107125 Numbers k such that (10^(2*k+1) + 36*10^k - 1)/9 is prime.

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Programs

Magma

Mathematica

PARI

Formula

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A107126 Numbers n such that (10^(2n+1)+45*10^n-1)/9 is prime.

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Mathematica

A107125 Numbers k such that (10^(2k+1) + 3610^k - 1)/9 is prime.

A077777 Numbers k such that 7(10^k - 1)/9 - 510^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

A077781 Numbers k such that 7(10^k - 1)/9 - 310^floor(k/2) is a palindromic wing prime (also known as near-repdigit palindromic prime).