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A305835 Primes which oscillate from prime to composite under a cyclic shift of digits. 1

%I #35 Jul 03 2018 10:10:48

%S 19,23,29,41,43,47,53,59,61,67,83,89,1163,1321,1361,1783,1933,1997,

%T 2113,2161,2333,2339,2347,2381,2389,2393,2729,2741,2777,2927,2963,

%U 2999,3319,3323,3347,3389,3391,3923,4127,4157,4349,4357,4363,4397,4723,4733,4751,4787,4943,4957,4969,4973,4999

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

%C 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.

%C All terms have even length.

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

%H Philip Mizzi, <a href="/A305835/b305835.txt">Table of n, a(n) for n = 1..101</a>

%e n=1

%e N_0 = 19 (prime)

%e N_1 = 91 (nonprime)

%e N_2 = N_0 = 19 (prime)

%e .

%e .

%e n=13 [left cyclic shift]

%e N_0 = 1163 (prime)

%e N_1 = 1631 (nonprime)

%e N_2 = 6311 (prime)

%e N_3 = 3116 (nonprime)

%e N_4 = N_0 = 1163 (prime)

%e .

%e .

%e n=13 [right cyclic shift]

%e N_0 = 1163 (prime)

%e N_1 = 3116 (nonprime)

%e N_2 = 6311 (prime)

%e N_3 = 1631 (nonprime)

%e N_4 = N_0 = 1163 (prime)

%t 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 *)

%o (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

%Y Cf. A286415 (provides the first terms only).

%K nonn,base

%O 1,1

%A _Philip Mizzi_, Jun 11 2018

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