A077798 -id:A077798 - cp's OEIS frontend

A183181 Numbers k such that (710^(2k+1) - 9*10^k - 7)/9 is prime.

4, 5, 8, 11, 1244, 1685, 2009, 14657, 15118, 20332, 50830, 75062

Offset: 1

Ray Chandler, Dec 28 2010

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...77677...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) - 9*10^n - 7)/9], Print[n]], {n, 3000}]

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

a(n) = (A077788(n) - 1)/2.

a(10) from Robert Price, Oct 07 2023
a(11) from Robert Price, Oct 17 2023
a(12) from Robert Price, Dec 06 2023

A183182 Numbers k such that (710^(2k+1) + 9*10^k - 7)/9 is prime.

Original entry on oeis.org

1, 3, 39, 54, 168, 240, 5328, 6159, 24675, 52227, 113887

Offset: 1

Ray Chandler, Dec 28 2010

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...77877...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) + 9*10^n - 7)/9], Print[n]], {n, 3000}]

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

a(n) = (A077793(n) - 1)/2.

a(9) from Robert Price, Oct 07 2023
a(10) from Robert Price, Oct 30 2023
a(11) from Robert Price, Aug 03 2024

A183183 Numbers n such that (710^(2n+1)+1810^n-7)/9 is prime.

Original entry on oeis.org

1, 2, 8, 19, 20, 212, 280, 887, 1021, 5515, 8116, 11852

Offset: 1

Ray Chandler, Dec 28 2010

a(13) > 10^5. - Robert Price, Jan 19 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 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^(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) = (A077796(n)-1)/2.

A183185 Numbers n such that 10^(2n+1)-5*10^n-1 is prime.

Original entry on oeis.org

14, 22, 36, 104, 1136, 17864, 25448

Offset: 1

Ray Chandler, Dec 28 2010

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^(2n + 1) - 5*10^n - 1], Print[n]], {n, 3000}]

is(n)=ispseudoprime(10^(2*n+1)-5*10^n-1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077782(n)-1)/2.

A183186 Numbers k such that 10^(2k+1) - 4*10^k - 1 is prime.

Original entry on oeis.org

88, 112, 198, 622, 4228, 10052, 55862

Offset: 1

Ray Chandler, Dec 28 2010

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...99599...99.
Index entries for primes involving repunits.

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

Do[If[PrimeQ[10^(2n + 1) - 4*10^n - 1], Print[n]], {n, 3000}]

is(n)=ispseudoprime(10^(2*n+1)-4*10^n-1) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = (A077786(n) - 1)/2.

A372141 Primes p that are palindromic in some prime base q, where q < p.

Original entry on oeis.org

3, 5, 7, 13, 17, 23, 31, 41, 67, 71, 73, 83, 107, 109, 127, 151, 157, 173, 199, 233, 257, 271, 277, 307, 313, 353, 379, 409, 419, 421, 431, 443, 457, 499, 521, 523, 571, 587, 599, 601, 631, 643, 647, 653, 691, 701, 709, 719, 733, 743, 757, 787, 797, 809, 823, 829, 857, 863, 887

Offset: 1

Tadayoshi Kamegai, Apr 20 2024

If we remove either constraint of q < p or q being prime, then the sequence would be all prime numbers (A000040).
By definition it is a superset of A016041, and is a proper superset by construction (e.g., 13 is in the sequence).
Some terms have multiple bases that yield palindromic representations, the first being 31 (which is palindromic in both base 2 and base 5). The smallest prime p such that there exist n distinct primes less than p that give palindromic representations of p is A372142(n).

			11 is not in this sequence as its representation in base 2 is 1011, in base 3 is 102, in base 5 is 21, in base 7 is 14, none of which are palindromic.
1483 is in this sequence as its representation in base 37 is 131, which is palindromic.

Chai Wah Wu, Table of n, a(n) for n = 1..11371 (terms 1..1000 from Tadayoshi Kamegai)

Cf. A372142, A002385, A002113.

Cf. A007500, A016041, A077798.

a={}; For[i=1, i<=155, i++, flag=0; For[j=1, Prime[j] < Prime[i] && flag==0, j++, If[PalindromeQ[IntegerDigits[Prime[i], Prime[j]]], flag=1; AppendTo[a, Prime[i]]]]]; a (* Stefano Spezia, Apr 22 2024 *)

from sympy import sieve
from sympy.ntheory import digits
from itertools import islice
def ispal(v): return v == v[::-1]
def agen(): yield from (p for p in sieve if any(ispal(digits(p, q)[1:]) for q in sieve.primerange(1, p)))
print(list(islice(agen(), 60))) # Michael S. Branicky, Apr 20 2024

A077787 Numbers k such that (10^k - 1)/9 + 5*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

21, 29, 81, 119, 321, 825, 1121, 2579, 3693

Offset: 1

Patrick De Geest, Nov 16 2002

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

a(10) > 4*10^5. - _Robert Price, Jan 23 2025

			21 is a term because (10^21 - 1)/9 + 5*10^10 = 111111111161111111111.

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^n + 45*10^Floor[n/2] - 1)/9], Print[n]], {n, 3, 4000, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A107126(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

A372142 a(n) is the smallest prime p such that there exist exactly n distinct primes q where q < p and the representation of p in base q is a palindrome.

Original entry on oeis.org

2, 3, 31, 443, 23053, 86677, 11827763, 27362989, 755827199, 1306369439

Offset: 0

Tadayoshi Kamegai, Apr 21 2024

This is a special case of A372141.
It need not be the case that a(n) is a palindrome in base 2, as 23053 is a counterexample.
For p > 3, one only needs to check q such that q^2 + 1 <= p else p = cc_q = c*(q+1), not prime for c != 1 and q != 2. A similar argument shows that p cannot have an even number of digits in base q, else it would be divisible by (q+1). - Michael S. Branicky, Apr 21 2024

			a(5) = 86677, as it is palindromic in base 2, 107, 113, 151, and 233, and no smaller number satisfies the property.

Cf. A372141, A002385, A002113.

Cf. A016041, A007500, A077798.

from math import isqrt
from sympy import sieve
from sympy.ntheory import digits
from itertools import islice
def ispal(v): return v == v[::-1]
def f(p): return sum(1 for q in sieve.primerange(1, isqrt(p-1)+1) if ispal(digits(p, q)[1:]))
def agen():
    adict, n = {0:2, 1:3}, 0
    for p in sieve:
        v = f(p)
        if v >= n and v not in adict:
            adict[v] = p
            while n in adict:
                yield adict[n]; del adict[n]; n += 1
print(list(islice(agen(), 6))) # Michael S. Branicky, Apr 21 2024

a(6) from Jon E. Schoenfield, Apr 21 2024
a(7) from Michael S. Branicky, Apr 21 2024
a(8) from Michael S. Branicky, Apr 22 2024
a(9) from Michael S. Branicky, Apr 24 2024

A320516 Palindromic wing primes that are also Lychrel candidates.

Original entry on oeis.org

7774777, 777767777, 77777677777, 99999199999, 1111118111111, 7777774777777, 111111181111111, 333333373333333, 77777777677777777, 99999999299999999, 9999999992999999999, 33333333333733333333333, 77777777777677777777777, 333333333333373333333333333

Offset: 1

Robert James Liguori, Oct 29 2018

Lychrel candidates are natural numbers that seem unable to form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers.

On January 23, 2017 a Russian schoolboy, Andrey S. Shchebetov, announced on his web site that he had found a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. That sequence was published in the OEIS as A281506. The trajectory of the last number of that sequence, 1186061987030929990, under the "Reverse and Add!" operation was published separately in the OEIS as A281507.

Robert Liguori, Github project SequencePier

Cf. A077798, A000040, A023108, A281506, A281507.

Seven terms inserted by Jon E. Schoenfield, Oct 31 2018

a(14) from Jon E. Schoenfield, Nov 01 2018

cp's OEIS Frontend

A183181 Numbers k such that (7*10^(2*k+1) - 9*10^k - 7)/9 is prime.

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A183182 Numbers k such that (7*10^(2*k+1) + 9*10^k - 7)/9 is prime.

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A183183 Numbers n such that (7*10^(2n+1)+18*10^n-7)/9 is prime.

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A183185 Numbers n such that 10^(2n+1)-5*10^n-1 is prime.

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A183186 Numbers k such that 10^(2k+1) - 4*10^k - 1 is prime.

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A372141 Primes p that are palindromic in some prime base q, where q < p.

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A077787 Numbers k such that (10^k - 1)/9 + 5*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A183181 Numbers k such that (710^(2k+1) - 9*10^k - 7)/9 is prime.

A183182 Numbers k such that (710^(2k+1) + 9*10^k - 7)/9 is prime.

A183183 Numbers n such that (710^(2n+1)+1810^n-7)/9 is prime.