A105992 -id:A105992 - cp's OEIS frontend

A088281 a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '1's.

11, 101, 131, 151, 181, 191, 11311, 11411, 1114111, 1117111, 111181111, 111191111, 1111118111111, 111111151111111, 111111181111111, 111111111161111111111, 11111111111111611111111111111, 111111111111111111131111111111111111111, 11111111111111111111111111911111111111111111111111111

Offset: 0

Amarnath Murthy, Sep 29 2003

For n > 1, near-repunit palindromic primes (or, palindromic terms of A105992). - Lekraj Beedassy, Jun 05 2009

Harvey P. Dale, Table of n, a(n) for n = 0..32
Index to OEIS entries related to primes involving repdigits.

Cf. A088282, A088283, A088284 (analog with string of '3's, '7's resp. '9's).
Cf. A105992 (near-repunit primes), A065074 (which contain the digit 0), A034093 (number of primes by changing one 1 to 0), A065083 (least k for which that = n).
Cf. A164937 (near-repdigit primes); with 2, ..., 9 as repeated digit: A105982, A105981, A105980, A105979, A105978, A105977, A105976, A105975.

Join[{11},Select[Flatten[Table[FromDigits[Join[PadRight[{},n,1],{d},PadRight[{},n,1]]],{n,26},{d,Cases[Range[0,9],Except[1]]}]],PrimeQ]] (* Harvey P. Dale, Nov 04 2024 *)

print1(11); for(L=1,19,for(d=0,9,d!=1 && ispseudoprime(p=10^(2*L+1)\9+(d-1)*10^L) && print1(","p))) \\ M. F. Hasler, Feb 07 2020

More terms from David Wasserman, Aug 03 2005
Offset changed from 0 to 1 by Lekraj Beedassy, Jun 05 2009
Edited by M. F. Hasler, Feb 07 2020

A164937 Near-repdigit primes.

Original entry on oeis.org

101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997, 1117, 1151, 1171, 1181, 1511

Offset: 1

G. L. Honaker, Jr., Aug 31 2009

Robert G. Wilson v, Table of n, a(n) for n = 1..22172 (first 5000 terms from Arkadiusz Wesolowski)
Chris Caldwell, The Top 20 Near-repdigit Primes
Chris Caldwell, The Prime Glossary, Near-repdigit prime
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015)

Cf. A105992, A105975, A105976, A105977, A105978, A105979, A105980, A105981, A105982, A065074.

lst = {}; Do[If[PrimeQ[n] && SortBy[Tally[IntegerDigits[n]], Last][[-1, -1]] == IntegerLength[n] - 1, AppendTo[lst, n]], {n, 101, 10^3}]; lst (* Arkadiusz Wesolowski, Sep 18 2011 *)
lst = {}; Do[r = (10^n - 1)/9; Do[AppendTo[lst, DeleteCases[Select[FromDigits[Permutations[Append[IntegerDigits[a*r], d]]], PrimeQ], r | 2 | 3 | 5 | 7]], {a, 9}, {d, 0, 9}], {n, 2, 6}]; Sort[Flatten[lst]] (* Arkadiusz Wesolowski, Sep 22 2011 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for d in count(3):
        ds = set()
        for end in "1379":
            ds.update(int(c*(d-1) + end) for c in "123456789" if c != end)
            for diff in "0123456789":
                if end == diff: continue
                cands = (end*i + diff + end*(d-1-i) for i in range(d-1))
                ds.update(int(t) for t in cands if t[0] != "0")
        yield from sorted(t for t in ds if isprime(t))
print(list(islice(agen(), 52))) # Michael S. Branicky, May 17 2022

Three more terms from Lekraj Beedassy, Dec 06 2009

A331860 Numbers k such that R(k) + 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

6, 7, 12, 31, 58, 127, 454, 556, 558, 604, 2944, 8118, 12078, 16942, 26268, 45198

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are near-repunit primes, cf. A105992.
In base 10, R(k) + 10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 2 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 2 just left of the middle of the repunit of length k.
No term can be congruent to 2 (mod 3). - Chai Wah Wu, Feb 07 2020

			For n = 6,  R(6)  + 10^(3-1) = 111211 is prime.
For n = 7,  R(7)  + 10^(3-1) = 1111211 is prime.
For n = 12, R(12) + 10^(6-1) = 111111211111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).

Cf. A331861 (variant with floor(n/2) instead of floor(n/2-1)), A331863 (variant with - (digit 0) instead of + (digit 2)).

for(n=2,999,isprime(p=10^n\9+10^(n\2-1))&&print1(n","))

a(8)-a(14) from Giovanni Resta, Jan 31 2020

a(15)-a(16) from Michael S. Branicky, Jul 23 2024

A182051 Primes with a majority of one digit.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997, 1117, 1151

Offset: 1

Arkadiusz Wesolowski, Apr 08 2012

a(n+5) = A164937(n) for n <= 89.

			1151 is prime and the number of ones is greater than the number of all other digits, so this number is in the sequence.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Supersequence of A164937 and of A164968.

Cf. A004022, A105992.

lst = {}; Do[i = IntegerDigits[n]; If[PrimeQ[n] && Count[i, First[Commonest@i]] > IntegerLength[n]/2, AppendTo[lst, n]], {n, 10^4}]; lst

A331861 Numbers n for which R(n) + 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

1, 6, 10, 18, 24, 4978

Offset: 1

M. F. Hasler, Jan 30 2020

The primes corresponding to the terms of the sequence are a subset of the near-repunit primes A105992.
In base 10, R(n) + 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 2, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 2 is just left to the middle of the number.
There cannot be an odd term > 1 since the corresponding palindrome factors as R((n+1)/2)*(10^((n-1)/2) + 1).
No term can be congruent to 2 mod 3. - Chai Wah Wu, Feb 07 2020

			For n = 1, R(n) + 10^floor(n/2) = 2 is prime.
For n = 6, R(n) + 10^floor(n/2) = 112111 is prime.
For n = 10, R(n) + 10^floor(n/2) = 1111211111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).

Cf. A331860 (variant with floor(n/2-1) instead of floor(n/2)), A331862 (variant with - (digit 0) instead of + (digit 2)).

for(n=0,9999,isprime(p=10^n\9+10^(n\2))&&print1(n","))

a(6) from Daniel Suteu, Feb 01 2020

A331862 Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

3, 26, 186, 206, 258, 3486, 12602

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
In base 10, R(n) - 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 0, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 0 is just left to the middle of the number.
There can't be an odd term > 3 because the corresponding palindrome factors as R((n-1)/2)*(10^((n+1)/2) + 1).
No term can be congruent to 1 mod 3. - Chai Wah Wu, Feb 07 2020

			For n = 3, R(n) - 10^floor(n/2) = 101 is prime.
For n = 26, R(n) - 10^floor(n/2) = 11111111111101111111111111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015)

Cf. A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A105992 (near-repunit primes), A138148 (Cyclops numbers with digits 0 & 1).

Cf. A331860 (variant with digit 2 instead of digit 0), A331863 (variant with floor(n/2-1) instead of floor(n/2)).

for(n=0,9999,isprime(p=10^n\9-10^(n\2))&&print1(n","))

a(6)-a(7) from Giovanni Resta, Jan 31 2020

A331864 Numbers k such that R(k) + 2*10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

2, 3, 5, 8, 9, 39, 78, 81, 155, 249, 387, 395, 510, 711, 1173, 1751, 10245

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are near-repunit primes, cf. A105992.
In base 10, R(k) + 2*10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 3 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 3 just left of the middle of the repunit of length k.
No term can be equivalent to 1 (mod 3). - Chai Wah Wu, Feb 07 2020

			For k = 2, R(2) + 2*10^(1-1) = 13 is prime.
For k = 3, R(3) + 2*10^(1-1) = 113 is prime.
For k = 5, R(5) + 2*10^(2-1) = 11131 is prime.
For k = 8, R(8) + 2*10^(4-1) = 11113111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A011557 (powers of 10).

Cf. A331865 (variant with floor(n/2) instead of floor(n/2-1)), A331860, A331863 (variants with digit 2 resp. 0 instead of digit 3).

for(n=2,999,isprime(p=10^n\9+2*10^(n\2-1))&&print1(n","))

a(13)-a(16) from Daniel Suteu, Feb 01 2020

a(17) from Michael S. Branicky, Feb 03 2023

A331865 Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 12, 20, 39, 74, 78, 80, 84, 104, 195, 654, 980, 2076, 5940, 19637

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 2*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 3, and again floor(n/2) digits 1 (except for n=0). For odd n, this is a palindrome (a.k.a. wing prime, cf. A077779), for even n the digit 3 is just left to the middle of the number.
a(22) > 50000. - Michael S. Branicky, Feb 19 2025

			For n = 0, R(0) + 2*10^floor(0/2) = 2 is prime.
For n = 1, R(1) + 2*10^floor(1/2) = 3 is prime.
For n = 2, R(2) + 2*10^floor(2/2) = 31 is prime.
For n = 3, R(3) + 2*10^floor(3/2) = 131 is prime.
For n = 5, R(5) + 2*10^floor(5/2) = 11311 is prime.
For n = 6, R(6) + 2*10^floor(6/2) = 113111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331860 & A331863 (variants with digit 2 resp. 0 instead of 3), A331864 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077779 (odd terms).

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 2*10^Floor[#/2]] &] (* Michael De Vlieger, Jan 31 2020 *)

for(n=0,9999,isprime(p=10^n\9+2*10^(n\2))&&print1(n","))

a(18)-a(20) from Giovanni Resta, Jan 30 2020

a(21) from Michael S. Branicky, Feb 19 2025

A331866 Numbers k for which R(k) + 3*10^floor(k/2) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

0, 2, 5, 7, 8, 10, 65, 91, 208, 376, 586, 2744, 3089, 19378, 20246

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subset of the near-repunit primes A105992 (at least when they have k > 2 digits).
In base 10, R(k) + 3*10^floor(k/2) has k digits all of which are 1 except for one digit 4 (for k > 0) located in the center (for odd k) or just to the left of it (for even k): i.e., there are ceiling(k/2)-1 digits 1 to the left and floor(k/2) digits 1 to the right of the digit 4. For odd k, this is a palindrome a.k.a. wing prime, cf. A077780, the subsequence of odd terms.
a(14) = 19378 was found by Amiram Eldar, verified to be the 14th term in collaboration with the author of the sequence and factordb.com. The term a(13) = 3089 corresponds to a certified prime (Ivan Panchenko, 2011, cf. factordb.com); a(12) and a(14) are only PRP as far as we know.

			For n = 0, R(0) + 3*10^floor(0/2) = 3 is prime.
For n = 2, R(2) + 3*10^floor(2/2) = 41 is prime.
For n = 5, R(5) + 3*10^floor(5/2) = 11411 is prime.
For n = 7, R(7) + 3*10^floor(7/2) = 1114111 is prime.
For n = 8, R(8) + 3*10^floor(8/2) = 11141111 is prime.

Chris Caldwell, The Prime Glossary, Near-repunit prime.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).
Cf. A331862, A331861, A331865, A331869 (variants with digit 0, 2, 3 or 5 instead of 4), A331867 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077780 (odd terms).

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 3*10^Floor[#/2]] &]

for(n=0,9999,ispseudoprime(p=10^n\9+3*10^(n\2))&&print1(n","))

a(15) from Michael S. Branicky, Sep 24 2024

A331867 Numbers n for which R(n) + 3*10^floor(n/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

Original entry on oeis.org

68, 5252, 5494, 7102

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 3*10^floor(n/2-1) has ceiling(n/2) digits 1, one digit 4, and again floor(n/2-1) digits 1. For odd and even n, the digit 4 is just to the right of the middle of the number.
For odd n = 2m + 1, f(n) = R(n) + 3*10^floor(n/2-1) is divisible by 3, 7 or 13 when m is congruent 1 or 4, 3 or 5, resp. 0 or 2 (mod 6): there can't be an odd term.
For even n = 2m, f(n) is divisible by 3 or 7 when m is congruent to 0 or 3, resp. 1 or 2 (mod 6). When m = 6k + 4, then f(n) is prime for k = 5 and 437 (and no further k <= 600), and divisible by 23 or 53 when k is congruent to 10 (mod 11) resp. 3 (mod 13). When m = 6k + 5, f(n) is prime for k = 457 and 591 and no other value up to 600, and divisible by 23, 47, 53, 97, 163, 181, 859, ... for k congruent to 5 (mod 11), 11 (mod 23), 5 (mod 13), 0 (mod 32), 13 (mod 27), 26 (mod 30), 3 (mod 13), ..., respectively.
a(5) > 7272.

			For n = 2, R(2) + 3*10^floor(2/2-1) = 14 = 2*7 is not prime.
For n = 3, R(3) + 3*10^floor(3/2-1) = 114 = 2*3*19 is not prime.
For n = 4, R(4) + 3*10^floor(4/2) = 1141 = 7*163 is not prime.
For n = 5, R(5) + 3*10^floor(5/2) = 11141 = 13*857 is not prime.
For n = 68, R(68) + 3*10^floor(68/2) = 1...1141...1 is prime, with 34 digits '1' to the left of a digit '4' and 33 digits '1' to its right.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Index to OEIS entries related to primes involving repdigits.

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).

Cf. A331860, A331863, A331864 (variants with digit 2, 0 resp. 3 instead of 4), A331866 (variant with floor(n/2) instead of floor(n/2-1)).

Select[Range[2, 2500], PrimeQ[(10^# - 1)/9 + 3*10^Floor[#/2 - 1]] &]  (* corrected by Amiram Eldar, Feb 10 2020 *)

for(n=2,9999,isprime(p=10^n\9+3*10^(n\2-1))&&print1(n","))

cp's OEIS Frontend

A088281 a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '1's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A164937 Near-repdigit primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A331860 Numbers k such that R(k) + 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A182051 Primes with a majority of one digit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A331861 Numbers n for which R(n) + 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A331862 Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A331864 Numbers k such that R(k) + 2*10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A331865 Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Views

Author

Keywords