A331866 -id:A331866 - 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.

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","))

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

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

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