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

1,1

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

Links

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

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A286415 (provides the first terms only).

Sequence in context: A076056 A068654 A286415 * A019384 A088987 A364458

Adjacent sequences: A305832 A305833 A305834 * A305836 A305837 A305838

Keyword

nonn,base

Author

Philip Mizzi, Jun 11 2018

Status

approved