A183176 -id:A183176 - cp's OEIS frontend

A077790 Numbers k such that (10^k - 1)/3 + 4*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

3, 7, 15, 23, 27, 35, 59, 63, 67, 155, 1867, 3111, 23517, 235415

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.

a(14) > 200000. - Robert Price, Dec 29 2016

			23 is a term because (10^23 - 1)/3 + 4*10^11 = 33333333333733333333333.

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 33...33733...33
Index entries for primes involving repunits.

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

Do[ If[ PrimeQ[(10^n + 12*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 23600, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183176(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

a(14) from Robert Price, Oct 30 2023

A332137 a(n) = (10^(2n+1)-1)/3 + 4*10^n.

Original entry on oeis.org

7, 373, 33733, 3337333, 333373333, 33333733333, 3333337333333, 333333373333333, 33333333733333333, 3333333337333333333, 333333333373333333333, 33333333333733333333333, 3333333333337333333333333, 333333333333373333333333333, 33333333333333733333333333333, 3333333333333337333333333333333

Offset: 0

M. F. Hasler, Feb 09 2020

See A183176 = {1, 3, 7, 11, 13, 17, 29, 31, ...} for the indices of primes.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).
Patrick De Geest, Palindromic Wing Primes: (3)7(3), updated: June 25, 2017.
Makoto Kamada, Factorization of 33...33733...33, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077790-1)/2 = A183176: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332127 .. A332197 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

A332137 := n -> (10^(2*n+1)-1)/3+4*10^n;

Array[ (10^(2 # + 1)-1)/3 + 4*10^# &, 15, 0]

apply( {A332137(n)=10^(n*2+1)\3+4*10^n}, [0..15])

def A332137(n): return 10**(n*2+1)//3+4*10**n

a(n) = 3*A138148(n) + 7*10^n = A002277(2n+1) + 4*10^n.
G.f.: (7 - 404*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.

cp's OEIS Frontend

A077790 Numbers k such that (10^k - 1)/3 + 4*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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