A077780 -id:A077780 - 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.

A107124 Numbers k such that (10^(2k+1)+2710^k-1)/9 is prime.

Original entry on oeis.org

2, 3, 32, 45, 1544

Offset: 1

Farideh Firoozbakht, May 19 2005

k is in the sequence iff the palindromic number 1(k).4.1(k) is prime (dot between numbers means concatenation). If k is in the sequence then k is not of the forms 3m+1, 16m+11, 16m+12, 18m+11, 18m+15, etc. (the proof is easy).

			32 is in the sequence because the palindromic number (10^(2*32+1)+27*10^32-1)/9 = 1(32).4.1(32) =
11111111111111111111111111111111411111111111111111111111111111111 is prime.

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...11411...11
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 27*10^n - 1)/9], Print[n]], {n, 2200}]
Select[Range[1600],PrimeQ[FromDigits[Join[PadRight[{},#,1],{4},PadRight[ {},#,1]]]]&] (* Harvey P. Dale, Aug 01 2017 *)

is(n)=ispseudoprime((10^(2*n+1)+27*10^n-1)/9) \\ Charles R Greathouse IV, May 22 2017

a(n) = (A077780(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010

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

cp's OEIS Frontend

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

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A107124 Numbers k such that (10^(2k+1)+2710^k-1)/9 is prime.

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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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cp's OEIS Frontend

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

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A107124 Numbers k such that (10^(2*k+1)+27*10^k-1)/9 is prime.

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Extensions

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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A107124 Numbers k such that (10^(2k+1)+2710^k-1)/9 is prime.