A077791 -id:A077791 - cp's OEIS frontend

A107648 Numbers n such that (10^(2n+1)+63*10^n-1)/9 is prime.

1, 4, 6, 7, 384, 666, 675, 3165, 131020

Offset: 1

Farideh Firoozbakht, May 19 2005

n is in the sequence iff the palindromic number 1(n).8.1(n) is prime (dot between numbers means concatenation). Let f(n)=(10^(2n+1)+63*10^n-1)/9 then for all nonnegative integers m we have: I. 3 divides f(3m+2) II. 19 divides f(18m+13) III. 29 divides f(28*m+16) & 29 divides f(28*m+25) IV. 31 divides f(30*m+2) & 31 divides f(30*m+17) V. 41 divides f(5m+3), etc. So if n is in the sequence then n is not of the forms 3m+2, 18m+13, 28m+16 28m+25, 30m+2, 30m+17, 5m+3, etc.

a(9) > 10^5. - Robert Price, Oct 30 2017

			7 is in the sequence because (10^15+63*10^7-1)/9=1(7).8.1(7)=111111181111111 is prime.
666 is in the sequence because (10^(2*666+1)+63*10^666-1)/9=1(666).8.1(666) is prime.

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 141.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 11...11811...11
Index entries for primes involving repunits.

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

Do[If[PrimeQ[(10^(2n + 1) + 63*10^n - 1)/9], Print[n]], {n, 4000}]

for(n=0,1e4,if(ispseudoprime(t=(10^(2*n+1)+63*10^n)\9),print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = (A077791(n)-1)/2.

Edited by Ray Chandler, Dec 28 2010

a(9) from Robert Price, Aug 03 2024

A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.

Original entry on oeis.org

8, 181, 11811, 1118111, 111181111, 11111811111, 1111118111111, 111111181111111, 11111111811111111, 1111111118111111111, 111111111181111111111, 11111111111811111111111, 1111111111118111111111111, 111111111111181111111111111, 11111111111111811111111111111, 1111111111111118111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

See A107648 = {1, 4, 6, 7, 384, 666, ...} 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)8(1), updated: June 25, 2017.
Makoto Kamada, Factorization of 11...11811...11, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077791-1)/2 = A107648: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes), A077798 (palindromic wing primes), A088281 (primes 1..1x1..1), A068160 (smallest of given length), A053701 (vertically symmetric numbers).
Cf. A332128 .. A332178, A181965 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

A332118 := n -> (10^(2*n+1)-1)/9+7*10^n;

Array[(10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]

apply( {A332118(n)=10^(n*2+1)\9+7*10^n}, [0..15])

def A332118(n): return 10**(n*2+1)//9+7*10**n

a(n) = A138148(n) + 8*10^n = A002275(2n+1) + 7*10^n.
G.f.: (8 - 707*x + 600*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

A107648 Numbers n such that (10^(2n+1)+63*10^n-1)/9 is prime.

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