cp's OEIS Frontend

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

%I A305835 #39 Jun 18 2025 07:31:32
%S A305835 19,23,29,41,43,47,53,59,61,67,83,89,1163,1321,1361,1783,1933,1997,
%T A305835 2113,2161,2333,2339,2347,2381,2389,2393,2729,2741,2777,2927,2963,
%U A305835 2999,3319,3323,3347,3389,3391,3923,4127,4157,4349,4357,4363,4397,4723,4733,4751,4787,4943,4957,4969,4973,4999
%N A305835 Primes which oscillate from prime to composite under a cyclic shift of digits.
%C A305835 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 A305835 All terms have even length.
%C A305835 The smallest terms with 6, 8,..., 18 digits are 112927, 11117363, 1111319791, 111111335143, 11112333396319, 1111115783474981, and 111111119937131947, respectively. - _Giovanni Resta_, Jun 13 2018
%H A305835 Philip Mizzi, <a href="/A305835/b305835.txt">Table of n, a(n) for n = 1..101</a>
%e A305835 n=1
%e A305835 N_0 = 19 (prime)
%e A305835 N_1 = 91 (nonprime)
%e A305835 N_2 = N_0 = 19 (prime)
%e A305835 .
%e A305835 .
%e A305835 n=13 [left cyclic shift]
%e A305835 N_0 = 1163 (prime)
%e A305835 N_1 = 1631 (nonprime)
%e A305835 N_2 = 6311 (prime)
%e A305835 N_3 = 3116 (nonprime)
%e A305835 N_4 = N_0 = 1163 (prime)
%e A305835 .
%e A305835 .
%e A305835 n=13 [right cyclic shift]
%e A305835 N_0 = 1163 (prime)
%e A305835 N_1 = 3116 (nonprime)
%e A305835 N_2 = 6311 (prime)
%e A305835 N_3 = 1631 (nonprime)
%e A305835 N_4 = N_0 = 1163 (prime)
%t A305835 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 A305835 (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 A305835 Cf. A286415 (provides the first terms only).
%K A305835 nonn,base
%O A305835 1,1
%A A305835 _Philip Mizzi_, Jun 11 2018