A305835 Primes which oscillate from prime to composite under a cyclic shift of digits.

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 1163, 1321, 1361, 1783, 1933, 1997, 2113, 2161, 2333, 2339, 2347, 2381, 2389, 2393, 2729, 2741, 2777, 2927, 2963, 2999, 3319, 3323, 3347, 3389, 3391, 3923, 4127, 4157, 4349, 4357, 4363, 4397, 4723, 4733, 4751, 4787, 4943, 4957, 4969, 4973, 4999

Offset: 1

Author

Philip Mizzi, Jun 11 2018

Keywords

• nonn
• base

Comments

Numbers with a zero digit have been excluded as cycling through these numbers would generate leading zeros, which is problematic as you continue to cycle.

All terms have even length.

The smallest terms with 6, 8,..., 18 digits are 112927, 11117363, 1111319791, 111111335143, 11112333396319, 1111115783474981, and 111111119937131947, respectively. - Giovanni Resta, Jun 13 2018

Examples

			n=1
N_0 = 19 (prime)
N_1 = 91 (nonprime)
N_2 = N_0 = 19 (prime)
.
.
n=13 [left cyclic shift]
N_0 = 1163 (prime)
N_1 = 1631 (nonprime)
N_2 = 6311 (prime)
N_3 = 3116 (nonprime)
N_4 = N_0 = 1163 (prime)
.
.
n=13 [right cyclic shift]
N_0 = 1163 (prime)
N_1 = 3116 (nonprime)
N_2 = 6311 (prime)
N_3 = 1631 (nonprime)
N_4 = N_0 = 1163 (prime)
		

Links

• Philip Mizzi, Table of n, a(n) for n = 1..101

Crossrefs

Cf. A286415 (provides the first terms only).

Programs

Mathematica

ok[n_] := Catch[Block[{d = IntegerDigits[n]}, If[Min[d] == 0 || OddQ[ Length[d]], Throw@ False]; Do[If[PrimeQ[ FromDigits@ RotateLeft[d, j]] == OddQ[j], Throw@ False], {j, Length[d]-1}]; True]]; Select[ Prime@ Range@ 670, ok] (* Giovanni Resta, Jun 12 2018 *)

PARI

ok(p)={my(k=logint(p,10)); k%2 && isprime(p) && vecmin(digits(p))>0 && !#select(i->isprime(p\10^i + p%10^i*10^(k+1-i))==i%2, [1..k])} \\ Andrew Howroyd, Jun 11 2018