A183177 -id:A183177

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A077792

Numbers k such that (10^k - 1)/3 + 5*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

+10
2

3, 15, 171, 189, 547, 713, 2155, 3595, 13517, 60465

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

a(11) > 2*10^5. - Robert Price, Apr 21 2016

REFERENCES

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

LINKS

Table of n, a(n) for n=1..10.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)

Makoto Kamada, Prime numbers of the form 33...33833...33

Index entries for primes involving repunits.

FORMULA

a(n) = 2*A183177(n) + 1.

EXAMPLE

15 is a term because (10^15 - 1)/3 + 5*10^7 = 333333383333333.

MATHEMATICA

Do[ If[ PrimeQ[(10^n + 15*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 13600, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

CROSSREFS

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

KEYWORD

more,nonn,base

AUTHOR

Patrick De Geest, Nov 16 2002

EXTENSIONS

a(10) from Robert Price, Apr 21 2016

Name corrected by Jon E. Schoenfield, Oct 31 2018

STATUS

approved

A332138

a(n) = (10^(2*n+1)-1)/3 + 5*10^n.

+10
2

8, 383, 33833, 3338333, 333383333, 33333833333, 3333338333333, 333333383333333, 33333333833333333, 3333333338333333333, 333333333383333333333, 33333333333833333333333, 3333333333338333333333333, 333333333333383333333333333, 33333333333333833333333333333, 3333333333333338333333333333333

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

See A183177 = {1, 7, 85, 94, 273, 356, ...} for the indices of primes.

LINKS

Table of n, a(n) for n=0..15.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Patrick De Geest, Palindromic Wing Primes: (3)8(3), updated: June 25, 2017.

Makoto Kamada, Factorization of 33...33833...33, updated Dec 11 2018.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

FORMULA

a(n) = 3*A138148(n) + 8*10^n = A002277(2n+1) + 5*10^n.

G.f.: (8 - 505*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.

MAPLE

A332138 := n -> (10^(2*n+1)-1)/3+5*10^n;

MATHEMATICA

Array[ (10^(2 # + 1)-1)/3 + 5*10^# &, 15, 0]

PROG

(PARI) apply( {A332138(n)=10^(n*2+1)\3+5*10^n}, [0..15])

(Python) def A332138(n): return 10**(n*2+1)//3+5*10**n

CROSSREFS

Cf. (A077792-1)/2 = A183177: 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. A332118 .. A332178, A181965 (variants with different repeated digit 1, ..., 9).

Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

KEYWORD

nonn,base,easy

AUTHOR

M. F. Hasler, Feb 09 2020

STATUS

approved

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