A331865 -id:A331865 - cp's OEIS frontend

A332113 a(n) = (10^(2n+1)-1)/9 + 2*10^n.

3, 131, 11311, 1113111, 111131111, 11111311111, 1111113111111, 111111131111111, 11111111311111111, 1111111113111111111, 111111111131111111111, 11111111111311111111111, 1111111111113111111111111, 111111111111131111111111111, 11111111111111311111111111111, 1111111111111113111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107123 = {0, 1, 2, 19, 97, 9818, ...} 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)3(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11311...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077779-1)/2 = A107123: indices of primes; A331864 & A331865 (non-palindromic variants).
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332123 .. A332193 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332113 := n -> (10^(2*n+1)-1)/9+2*10^n;

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

apply( {A332113(n)=10^(n*2+1)\9+2*10^n}, [0..15])

def A332113(n): return 10**(n*2+1)//9+2*10**n

a(n) = A138148(n) + 3*10^n = A002275(2n+1) + 2*10^n.
G.f.: (3 - 202*x + 100*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.

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

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

A331869 Numbers n for which R(n) + 4*10^floor(n/2) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

Original entry on oeis.org

1, 3, 4, 15, 76, 91, 231, 1363, 1714, 1942, 2497, 4963, 5379, 12397, 23224, 26395

Offset: 1

M. F. Hasler, Feb 09 2020

For n > 1, the corresponding primes are a subset of A105992: near-repunit primes.
In base 10, R(n) + 4*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 5, and again floor(n/2) digits 1, except for n = 0. For odd n, this is a palindrome (a.k.a. wing prime, cf. A077783: subsequence of odd terms), for even n the digit 5 is just left to the middle of the number.
See also the variant A331868 where the digit 5 is just to the right of the middle.

			For n = 1, R(1) + 4*10^floor(1/2) = 5 is prime.
For n = 3, R(3) + 4*10^floor(3/2) = 151 is prime.
For n = 4, R(4) + 4*10^floor(4/2) = 1511 is prime.
For n = 15, R(15) + 4*10^floor(15/2) = 111111151111111 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. A331862, A331861, A331865, A331866 (variants with digit 0, 2, 3 or 4 instead of 5), A331868 (variant with floor(n/2-1) instead of floor(n/2)).
Cf. A077783 (odd terms).

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 4*10^Floor[#/2]] &]

for(n=0,9999,ispseudoprime(p=10^n\9+4*10^(n\2))&&print1(n","))

a(12)-a(14) from Michael S. Branicky, Feb 03 2023

a(15)-a(16) from Michael S. Branicky, Apr 11 2023

cp's OEIS Frontend

A332113 a(n) = (10^(2n+1)-1)/9 + 2*10^n.

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

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

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A331869 Numbers n for which R(n) + 4*10^floor(n/2) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

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Mathematica

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