A077783 -id:A077783 - cp's OEIS frontend

A332115 a(n) = (10^(2n+1)-1)/9 + 4*10^n.

5, 151, 11511, 1115111, 111151111, 11111511111, 1111115111111, 111111151111111, 11111111511111111, 1111111115111111111, 111111111151111111111, 11111111111511111111111, 1111111111115111111111111, 111111111111151111111111111, 11111111111111511111111111111, 1111111111111115111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107125 = {0, 1, 7, 45, 115, 681, 1248, ...} 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)5(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11511...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077783-1)/2 = A107125: indices of primes; A331868 & A331869 (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. A332125 .. A332195 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332115 := n -> (10^(2*n+1)-1)/9+4*10^n;

Array[(10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]

apply( {A332115(n)=10^(n*2+1)\9+4*10^n}, [0..15])

def A332115(n): return 10**(n*2+1)//9+4*10**n

a(n) = A138148(n) + 5*10^n = A002275(2n+1) + 4*10^n.
G.f.: (5 - 404*x + 300*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.

A107125 Numbers k such that (10^(2k+1) + 3610^k - 1)/9 is prime.

Original entry on oeis.org

0, 1, 7, 45, 115, 681, 1248, 2481, 2689, 6198, 13197, 60126, 100072

Offset: 1

Farideh Firoozbakht, May 19 2005

k is in the sequence iff the palindromic number 1(k).5.1(k) is prime (dot between numbers means concatenation). If k is in the sequence then k is not of the forms 3m+2, 18m+12, 18m+14, 22m+4, 22m+6, etc. (the proof is easy).

a(14) > 100233. - _Robert Price, Sep 05 2023

			1248 is in the sequence because (10^(2*1248+1)+36*10^1248-1)/9=1(1248).5.1(1248) 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...11511...11
Index entries for primes involving repunits.

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

[n: n in [0..700] | IsPrime((10^(2*n+1)+36*10^n-1) div 9)]; // Vincenzo Librandi, Oct 13 2015

Do[If[PrimeQ[(10^(2n + 1) + 36*10^n - 1)/9], Print[n]], {n, 2200}]

is(n)=ispseudoprime((10^(2*n+1)+36*10^n-1)/9) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = (A077783(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010
a(12) from Robert Price, Oct 12 2015
a(13) from Robert Price, Sep 05 2023

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

A332115 a(n) = (10^(2n+1)-1)/9 + 4*10^n.

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A107125 Numbers k such that (10^(2*k+1) + 36*10^k - 1)/9 is prime.

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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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A107125 Numbers k such that (10^(2k+1) + 3610^k - 1)/9 is prime.