A287198 Numbers with the property that every cyclic permutation of its digits is a composite number with none of its permutations sharing any common prime factors.

4, 6, 8, 9, 25, 49, 52, 56, 58, 65, 85, 94, 116, 134, 145, 158, 161, 178, 187, 253, 275, 295, 325, 341, 358, 413, 451, 514, 527, 529, 532, 581, 583, 589, 611, 718, 752, 781, 815, 817, 835, 871, 895, 899, 952, 958, 989, 998, 1154, 1156, 1159, 1165, 1189, 1192

Offset: 1

Author

Luke Zieroth, May 21 2017

Keywords

• nonn
• base

Comments

Subsequence of A052382, as a number with a zero digit have cyclic permutations of the forms 0x and x0 which share prime factors of x. The only exception to this argument is 10, but 01 is not composite, so 10 is not a member of the sequence as well. - Chai Wah Wu, May 24 2017

If m is a multiple of 11 with an even number of digits, then m is not a term. - Chai Wah Wu, May 30 2017

Examples

			The numbers formed by cyclic permutations of 134 are 341 and 413. The factors of 134 are 2 and 67, the factors of 341 are 11 and 31, and the factors of 413 are 7 and 59. Since these numbers are all composite and none share any common factors with each other, 134 is included on the list.
		

Links

• Luke Zieroth and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 212 terms from Luke Zieroth)

Crossrefs

Cf. A052382, A068652, A067012.

Programs

Mathematica

ok[n_] := Catch@ Block[{t = FromDigits /@ (RotateLeft[IntegerDigits[n], #] & /@ Range[ IntegerLength@ n])}, If[! AllTrue[t, CompositeQ], Throw@False]; Do[ If[ GCD[t[[i]], t[[j]]] > 1, Throw@False], {i, Length@t}, {j, i-1}]; True]; Select[ Range@ 1200, ok] (* Giovanni Resta, May 25 2017 *)

PARI

is(n) = {my(d=digits(n), v=vector(#d)); v[1]=n; if(isprime(n)||n==10, return(0)); for(i=2, #d, v[i] = v[i-1]\10; v[i] = v[i]+(v[i-1]-v[i]*10)*10^(#d-1); if(isprime(v[i]), return(0)); for(j=1,i-1,if(gcd(v[j], v[i])>1, return(0)))); n>1} \\ David A. Corneth, May 25 2017

Python

from gmpy2 import is_prime, gcd, mpz
A287198_list, n  = [], 2
while n <= 10**6:
    s = str(n)
    if not is_prime(n) and '0' not in s:
        k = n
        for i in range(len(s)-1):
            s = s[1:]+s[0]
            m = mpz(s)
            if is_prime(m) or gcd(k,m) > 1:
                break
            k *= m
        else:
            A287198_list.append(n)
    n += 1 # Chai Wah Wu, May 27 2017