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A305885
Zeroless composite numbers which become and remain prime under a complete cyclic shift of digits.
1
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
1,1
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
LINKS
Robert Israel, Table of n, a(n) for n = 1..121 (first 56 terms from Philip Mizzi)
World of numbers, Circular prime
FORMULA
{ A052382 } intersection { A068653 }.
{ A068653 } minus { A011540 }.
EXAMPLE
N_0 = 1118 (composite);
N_1 = 1181 (prime);
N_2 = 1811 (prime);
N_3 = 8111 (prime);
N_4 = N_0 = 1118 (composite).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Philip Mizzi, Jun 13 2018
STATUS
approved