A331867 -id:A331867 - cp's OEIS frontend

A332114 a(n) = (10^(2n+1)-1)/9 + 3*10^n.

4, 141, 11411, 1114111, 111141111, 11111411111, 1111114111111, 111111141111111, 11111111411111111, 1111111114111111111, 111111111141111111111, 11111111111411111111111, 1111111111114111111111111, 111111111111141111111111111, 11111111111111411111111111111, 1111111111111114111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107124 = {2, 3, 32, 45, 1544, ...} 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)4(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11411...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077780-1)/2 = A107124: indices of primes; A331866 & A331867 (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. A332124 .. A332194 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332114 := n -> (10^(2*n+1)-1)/9+3*10^n;

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

apply( {A332114(n)=10^(n*2+1)\9+3*10^n}, [0..15])

def A332114(n): return 10**(n*2+1)//9+3*10**n

a(n) = A138148(n) + 4*10^n = A002275(2n+1) + 3*10^n.
G.f.: (4 - 303*x + 200*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.

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

A331868 Numbers k for which R(k) + 4*10^floor(k/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

Original entry on oeis.org

4, 147, 270, 1288, 1551, 3427

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A105992: near-repunit primes.
In base 10, R(n) + 4*10^floor(n/2-1) has ceiling(n/2) digits 1, one digit 5, and again floor(n/2-1) digits 1. For odd and even n as well, the digit 5 appears just to the right of the middle of the number.
a(7) > 10^4. - Daniel Suteu, Feb 10 2020
a(7) > 5*10^4. - Michael S. Branicky, Nov 02 2024

			For n = 4, R(4) + 4*10^floor(4/2-1) = 1151 is prime.
For n = 5, R(5) + 4*10^floor(5/2-1) = 11151 =  3^3*7*59 is not prime.
For n = 147, R(147) + 4*10^72 = 1(74)51(72) is prime, where (.) indicates how many times the preceding digit is repeated.

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. A331863, A331860, A331864, A331867 (variants with digit 0, 2, 3 resp. 4 instead of 5), A331869 (variant with floor(n/2) instead of floor(n/2-1)).

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

for(n=2,9999,isprime(p=10^n\9+4*10^(n\2-1))&&print1(n","))

a(6) from Daniel Suteu, Feb 10 2020

cp's OEIS Frontend

A332114 a(n) = (10^(2n+1)-1)/9 + 3*10^n.

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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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A331868 Numbers k for which R(k) + 4*10^floor(k/2-1) is prime, where R(n) = (10^n-1)/9 (repunit: A002275).

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Mathematica

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Extensions