A077779 -id:A077779

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A107123

Numbers n such that (10^(2n+1)+18*10^n-1)/9 is prime.

+10
45

0, 1, 2, 19, 97, 9818

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

OFFSET

1,3

COMMENTS

A number n is in the sequence iff the palindromic number 1(n).3.1(n) is prime (1(n) means n copies of 1; dot between numbers means concatenation). If n is a positive term of the sequence then n is not of the form 3m, 6m+4, 12m+10, 28m+5, 28m+8, etc. (the proof is easy).

The palindromic number 1(n).2.1(n) is never prime for n > 0 because it is (1.0(n-1).1)*(1(n+1)). - Robert Israel, Jun 11 2015

a(7) > 10^5. - Robert Price, Apr 02 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..6.

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

Makoto Kamada, Prime numbers of the form 11...11311...11

Index entries for primes involving repunits.

FORMULA

a(n) = (A077779(n-1)-1)/2, for n > 1. [Corrected by M. F. Hasler, Feb 06 2020]

EXAMPLE

19 is in the sequence because the palindromic number (10^(2*19+1)+18*10^19-1)/9 = 1(19).3.1(19) = 111111111111111111131111111111111111111 is prime.

MAPLE

select(n -> isprime((10^(2*n+1)+18*10^n-1)/9), [$0..100]); # Robert Israel, Jun 11 2015

MATHEMATICA

Do[If[PrimeQ[(10^(2n + 1) + 18*10^n - 1)/9], Print[n]], {n, 2500}]

PROG

(PARI) for(n=0, 1e4, if(ispseudoprime(t=(10^(2*n+1)+18*10^n)\9), print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

CROSSREFS

Cf. A004023, A077775-A077798, A107124, A107125, A107126, A107127, A107648, A107649, A115073, A183174-A183187.

KEYWORD

nonn,base,more

AUTHOR

Farideh Firoozbakht, May 19 2005

EXTENSIONS

Edited by Ray Chandler, Dec 28 2010

STATUS

approved

A332113

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

+10
11

3, 131, 11311, 1113111, 111131111, 11111311111, 1111113111111, 111111131111111, 11111111311111111, 1111111113111111111, 111111111131111111111, 11111111111311111111111, 1111111111113111111111111, 111111111111131111111111111, 11111111111111311111111111111, 1111111111111113111111111111111

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

OFFSET

0,1

COMMENTS

See A107123 = {0, 1, 2, 19, 97, 9818, ...} 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: (1)3(1), updated: June 25, 2017.

Makoto Kamada, Factorization of 11...11311...11, updated Dec 11 2018.

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

FORMULA

a(n) = A138148(n) + 3*10^n = A002275(2n+1) + 2*10^n.

G.f.: (3 - 202*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.

MAPLE

A332113 := n -> (10^(2*n+1)-1)/9+2*10^n;

MATHEMATICA

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

PROG

(PARI) apply( {A332113(n)=10^(n*2+1)\9+2*10^n}, [0..15])

(Python) def A332113(n): return 10**(n*2+1)//9+2*10**n

CROSSREFS

Cf. (A077779-1)/2 = A107123: indices of primes; A331864 & A331865 (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. A332123 .. A332193 (variants with different repeated digit 2, ..., 9).

Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

KEYWORD

nonn,base,easy

AUTHOR

M. F. Hasler, Feb 09 2020

STATUS

approved

A331865

Numbers n for which R(n) + 2*10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

+10
4

0, 1, 2, 3, 5, 6, 8, 12, 20, 39, 74, 78, 80, 84, 104, 195, 654, 980, 2076, 5940

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

OFFSET

1,3

COMMENTS

The corresponding primes are a subsequence of A105992: near-repunit primes.

In base 10, R(n) + 2*10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 3, and again floor(n/2) digits 1 (except for n=0). For odd n, this is a palindrome (a.k.a. wing prime, cf. A077779), for even n the digit 3 is just left to the middle of the number.

LINKS

Table of n, a(n) for n=1..20.

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

EXAMPLE

For n = 0, R(0) + 2*10^floor(0/2) = 2 is prime.

For n = 1, R(1) + 2*10^floor(1/2) = 3 is prime.

For n = 2, R(2) + 2*10^floor(2/2) = 31 is prime.

For n = 3, R(3) + 2*10^floor(3/2) = 131 is prime.

For n = 5, R(5) + 2*10^floor(5/2) = 11311 is prime.

For n = 6, R(6) + 2*10^floor(6/2) = 113111 is prime.

MATHEMATICA

Select[Range[0, 2500], PrimeQ[(10^# - 1)/9 + 2*10^Floor[#/2]] &] (* Michael De Vlieger, Jan 31 2020 *)

PROG

(PARI) for(n=0, 9999, isprime(p=10^n\9+2*10^(n\2))&&print1(n", "))

CROSSREFS

Cf. A105992 (near-repunit primes), A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10).

Cf. A331860 & A331863 (variants with digit 2 resp. 0 instead of 3), A331864 (variant with floor(n/2-1) instead of floor(n/2)).

Cf. A077779 (odd terms).

KEYWORD

nonn,base,hard,more

AUTHOR

M. F. Hasler, Jan 30 2020

EXTENSIONS

a(17)-a(19) from Giovanni Resta, Jan 30 2020

STATUS

approved

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