A305885 Zeroless composite numbers which become and remain prime under a complete cyclic shift of digits.

14, 16, 32, 34, 35, 38, 74, 76, 91, 92, 95, 98, 118, 119, 133, 176, 194, 316, 398, 712, 715, 731, 736, 772, 775, 778, 779, 793, 794, 914, 935, 973, 1118, 1195, 1312, 1336, 1774, 1937, 3112, 3199, 3337, 3379, 3394, 3772, 3992, 7132, 7198, 7318, 7376, 7771, 7912

Offset: 1

Author

Philip Mizzi, Jun 13 2018

Keywords

• nonn
• base

Comments

Numbers with a zero digit have been excluded as cycling through these numbers would generate leading zeros, which become problematic throughout the cycle.

3999131, 7919777, 37177739 and 391331191 are in this sequence, see the link. - Eric Chen, Jun 14 2018

The sequence contains all composites without the digit zero that can be obtained by cyclically permuting the digits of the terms of A270083. - Felix Fröhlich, Jul 10 2018

Examples

			N_0 = 1118 (composite);
N_1 = 1181 (prime);
N_2 = 1811 (prime);
N_3 = 8111 (prime);
N_4 = N_0 = 1118 (composite).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..121 (first 56 terms from Philip Mizzi)
• World of numbers, Circular prime

Crossrefs

Cf. A011540, A052382, A068653, A270083.

Programs

Maple

Q[1]:= [seq([i],i=1..9)]:
for d from 2 to 7 do Q[d]:= map(t -> seq([i,op(t)],i=1..9),Q[d-1]) od:
Res:= NULL: count:= 0:
for d from 2 to 7 do
  for q in Q[d] do P[q]:= isprime(add(q[i]*10^(i-1),i=1..d)) od;
  for q in Q[d] do
     if (not P[q]) and andmap(t -> P[ListTools:-Rotate(q,t)],[$1..d-1]) then
       count:= count+1;
       Res:= Res, add(q[i]*10^(i-1),i=1..d);
     fi
  od
od:
Res; # Robert Israel, Jul 10 2018

Mathematica

Select[Range[11, 8000], Function[{k, d}, And[CompositeQ@ k, FreeQ[d, 0], AllTrue[Array[FromDigits@ RotateLeft[d, #] &, IntegerLength@ k - 1], PrimeQ]]] @@ {#, IntegerDigits@ #} &] (* Michael De Vlieger, Jun 14 2018 *)

PARI

eva(n) = subst(Pol(n), x, 10)
rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
is(n) = my(d=digits(n), r=rot(d)); if(vecmin(d)==0, return(0), while(1, if(!ispseudoprime(eva(r)), return(0)); r=rot(r); if(r==d, return(1))))
forcomposite(c=1, 8000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Jul 10 2018

Formula

{ A052382 } intersection { A068653 }.

{ A068653 } minus { A011540 }.