A272232 Smallest k > 0 such that R_k//n//R_k is prime, where R_k is the repunit A002275(k) of length k and // denotes concatenation; or -1 if no such k exists.

1, 9, -1, 1, 2, 1, 10, 3, 1, 1, 3, -1, 2, 3, 33, 1, 2, 1, 1, 21, 1, 2, -1, 1, 7, 48, 292, 4, 3, 1, 1, 2, 1, -1, 135, -1, 1, -1, 1, 34, 3, 3, 40, 2, -1, 1, 3, 1, 1, 32, 61, 1, 2, 1, 137, -1, 3, 1, 2, 42, 1, 14, 1, 262, 2, 22, -1, 3, 9, 2, 33, 73, 1, 3, 1, 2, 3, -1, 2, 2, 1

Offset: 0

Author

Felix Fröhlich, Apr 23 2016

Keywords

• sign
• base

Comments

a(2) = -1 (see second comment in A258372).

a(n) = -1 if n > 0 is in A099814 (see fourth comment in A004022).

a(n) = -1 if n is of the form A000042(i)*10^j+A000042(i) for some j > i > 0, since the resulting number is divisible by A002275(k)//A000042(i).

a(n) = -1 if n is a term of A010785 with an even number of digits, since any number of the form 1..1d..d1..1 with an even number of digits d is divisible by 11.

a(n) = 1 if there exists an integer x such that n = (A002275(A004023(x))-A011557(x)-1)/10.

From Chai Wah Wu, Nov 07 2019: (Start)

a(n) = -1 if n has an even number of digits and is a multiple of 11. In particular, a(n) = -1 if n is a term of A056524.

a(n) = -1 if n = (10^k+1)(10^m-1)/9 for some m > 0, k >= 0.

(End)

a(140) > 20000. - Hans Havermann, May 21 2022

Examples

			a(0) = 1 since 101 is prime; a(1) refers to the prime 1111111111111111111.
a(124) = -1 because R_k//124//R_k is divisible by 125*10^k-1.
		

Links

• Hans Havermann, Table of n, a(n) for n = 0..139

Crossrefs

Cf. A000042, A002275, A056524, A099814, A004023, A258372, A306861.

Programs

Mathematica

Table[SelectFirst[Range[10^4], PrimeQ@ FromDigits@ Flatten@ {#, IntegerDigits@ n, #} &@ Table[1, {#}] &], {n, 0, 91}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Apr 25 2016, Version 10.2 *)

PARI

a(n) = my(k=1); while(!ispseudoprime(eval(Str((10^k-1)/9, n, (10^k-1)/9))), k++); k

Extensions

a(35)-a(80) from Giovanni Resta, May 01 2016

Escape clausae value changed to -1 by N. J. A. Sloane, May 17 2022