A077775 -id:A077775 - cp's OEIS frontend

A077794 Odd integers k such that 10^k - 1 - 10^((k-1)/2) is a prime of the form 9...989...9, called a palindromic wing prime or a near-repdigit palindromic prime.

53, 757, 2493, 3597, 5835, 46069, 95019, 104281, 134809

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.
The corresponding primes have a(n) digits all of which are '9's except for the middle digit which is an '8'. They are too large to be listed in a sequence on their own, cf. examples. See A077775-A077798 and A107123-A107127 for palindromic wing/near-repdigit primes with other digits. - M. F. Hasler, Mar 03 2019
1888529 is a term but its position is not known. - Jeppe Stig Nielsen, Jan 12 2024
a(10) > 600000. - Serge Batalov, Jan 17 2024

			a(1) = 53 corresponds to the 53-digit prime
  p = 99999999999999999999999999899999999999999999999999999.
a(2) = 757 corresponds to p = (10^757 - 1) - 10^378 = 99...99899...99.

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

Do[ If[ PrimeQ[10^n - 1*10^Floor[n/2] - 1], Print[n]], {n, 3, 104300, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

is(n)=bittest(n,0)&&ispseudoprime(10^n-1-10^(n\2))
forstep(n=1,oo,2,is(n)&&print1(n",")) \\ M. F. Hasler, Mar 03 2019

a(n) = 2*A183187(n) + 1.

a(9) from PWP table, added by Patrick De Geest, Nov 05 2014

A077795 Numbers k such that (10^k - 1)/9 + 8*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 9, 53, 375, 453, 1749, 26619, 68033, 85179

Offset: 1

Patrick De Geest, Nov 16 2002

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

a(10) > 400000. - Robert Price, Jan 30 2025

			9 is a term because (10^9 - 1)/9 + 8*10^4 = 111191111.

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.

a(n) = 2*A107649(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 30 2018

a(8)-a(9) from Robert Price, Jan 30 2025

A077796 Numbers k such that 7(10^k - 1)/9 + 210^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 5, 17, 39, 41, 425, 561, 1775, 2043, 11031, 16233, 23705

Offset: 1

Patrick De Geest, Nov 16 2002

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

a(13) > 2*10^5. - Robert Price, Jan 19 2016

			17 is a term because 7*(10^17 - 1)/9 + 2*10^8 = 77777777977777777.

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...77977...77
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(7*10^n + 18*10^Floor[n/2] - 7)/9], Print[n]], {n, 3, 23800, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183183(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

A077797 Numbers k for which there exist k-digit palindromic wing primes (a.k.a. near-repdigit palindromic primes) of the general form r(10^k - 1)/9 + (m-r)10^floor(k/2) where k is the number of digits (an odd number > 1), r is the repeated digit, and m (different from r) is the middle digit.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 35, 39, 41, 45, 47, 53, 59, 63, 65, 67, 73, 79, 81, 87, 91, 109, 117, 119, 123, 139, 155, 159, 171, 177, 181, 185, 189, 195, 209, 225, 231, 233, 237, 259, 321, 325, 337, 339, 355, 363, 371, 375, 397, 425, 453

Offset: 1

Patrick De Geest, Nov 16 2002

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

P. De Geest, PWP's Sorted By Length

Cf. A077775-A077798.

Name edited by Jon E. Schoenfield, Nov 04 2018

A183175 Numbers k such that (10^(2k+1) + 6*10^k - 1)/3 is prime.

Original entry on oeis.org

1, 2, 17, 79, 118, 162, 177, 185, 240, 824, 1820, 2354, 134811

Offset: 1

Ray Chandler, Dec 28 2010

a(13) > 10^5. - Robert Price, Apr 03 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...33533...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}]

is(n)=ispseudoprime((10^(2*n+1)+6*10^n-1)/3) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077784(n) - 1)/2.

a(13) from Robert Price, Aug 03 2024

A183176 Numbers k such that (10^(2k+1) + 12*10^k - 1)/3 is prime.

Original entry on oeis.org

1, 3, 7, 11, 13, 17, 29, 31, 33, 77, 933, 1555, 11758, 117707

Offset: 1

Ray Chandler, Dec 28 2010

a(14) > 100000. - Robert Price, Dec 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...33733...33.
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 12*10^n - 1)/3], Print[n]], {n, 3000}]

is(n)=ispseudoprime((10^(2*n+1)+12*10^n-1)/3) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077790(n) - 1)/2.

a(14) from Robert Price, Oct 30 2023

A183177 Numbers n such that (10^(2n+1)+15*10^n-1)/3 is prime.

Original entry on oeis.org

1, 7, 85, 94, 273, 356, 1077, 1797, 6758, 30232

Offset: 1

Ray Chandler, Dec 28 2010

a(11) > 10^5. - Robert Price, Apr 21 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...33833...33
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 15*10^n - 1)/3], Print[n]], {n, 3000}]

is(n)=ispseudoprime((10^(2*n+1)+15*10^n-1)/3) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077792(n)-1)/2.

a(10) from Robert Price, Apr 21 2016

A183178 Numbers k such that 7(10^(2k+1)-1)/9 - 5*10^k is prime.

Original entry on oeis.org

0, 1, 3, 7, 10, 12, 480, 949, 1945, 7548, 8923

Offset: 1

Ray Chandler, Dec 28 2010

Original name: Numbers n such that (7*10^(2n+1)-45*10^n-7)/9 is prime.

a(12) > 10^5. - Robert Price, Nov 02 2015

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^(2n + 1) - 45*10^n - 7)/9], Print[n]], {n, 3000}]

is(n)=ispseudoprime((7*10^(2*n+1)-45*10^n-7)/9) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077777(n-1)-1)/2 for n>1.

Name edited and a(1) = 0 inserted by M. F. Hasler, Feb 07 2020

A183179 Numbers n such that 7(10^(2n+1)-1)/9 - 310^n is prime.

Original entry on oeis.org

2, 3, 6, 23, 36, 69, 561, 723, 3438, 4104, 9020, 13977, 19655, 32400

Offset: 1

Ray Chandler, Dec 28 2010

Original name: Numbers n such that (7*10^(2n+1)-27*10^n-7)/9 is prime.

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

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^(2n + 1) - 27*10^n - 7)/9], Print[n]], {n, 3000}]

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

a(n) = (A077781(n)-1)/2.

a(14) from Robert Price, Nov 23 2015

A183180 Numbers k such that (710^(2k+1) - 1810^k - 7)/9 is prime.

Original entry on oeis.org

0, 1, 7, 13, 58, 129, 253, 1657, 2244, 2437, 7924, 9903, 11899, 18157, 18957, 23665, 105609

Offset: 1

Ray Chandler, Dec 28 2010

a(17) > 10^5. - Robert Price, Jun 23 2017

0 could be considered part of this sequence since the formula evaluates to 5 which is a degenerate form of the near-repdigit palindrome 777...77577...777 with 0 occurrences of the digit 7. - Robert Price, Jun 23 2017

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...77577...77
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(7*10^(2n + 1) - 18*10^n - 7)/9], Print[n]], {n, 3000}]

is(n)=ispseudoprime((7*10^(2*n+1)-18*10^n-7)/9) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077785(n) - 1)/2.

a(16) from Robert Price, Jun 23 2017

a(17) from Robert Price, Oct 12 2023

cp's OEIS Frontend

A077794 Odd integers k such that 10^k - 1 - 10^((k-1)/2) is a prime of the form 9...989...9, called a palindromic wing prime or a near-repdigit palindromic prime.

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A077795 Numbers k such that (10^k - 1)/9 + 8*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A077796 Numbers k such that 7*(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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A077797 Numbers k for which there exist k-digit palindromic wing primes (a.k.a. near-repdigit palindromic primes) of the general form r*(10^k - 1)/9 + (m-r)*10^floor(k/2) where k is the number of digits (an odd number > 1), r is the repeated digit, and m (different from r) is the middle digit.

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Author

Keywords

References

Links

Crossrefs

Extensions

A183175 Numbers k such that (10^(2k+1) + 6*10^k - 1)/3 is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A183176 Numbers k such that (10^(2k+1) + 12*10^k - 1)/3 is prime.

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Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A183177 Numbers n such that (10^(2n+1)+15*10^n-1)/3 is prime.

Views

Author

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Comments

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A077796 Numbers k such that 7(10^k - 1)/9 + 210^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

A077797 Numbers k for which there exist k-digit palindromic wing primes (a.k.a. near-repdigit palindromic primes) of the general form r(10^k - 1)/9 + (m-r)10^floor(k/2) where k is the number of digits (an odd number > 1), r is the repeated digit, and m (different from r) is the middle digit.

A183178 Numbers k such that 7(10^(2k+1)-1)/9 - 5*10^k is prime.

A183179 Numbers n such that 7(10^(2n+1)-1)/9 - 310^n is prime.

A183180 Numbers k such that (710^(2k+1) - 1810^k - 7)/9 is prime.