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

1,1

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A052382, A068652, A067012.

Sequence in context: A046351 A161732 A066307 * A099071 A156673 A073866

Adjacent sequences: A287195 A287196 A287197 * A287199 A287200 A287201

Keyword

nonn,base

Author

Luke Zieroth, May 21 2017

Status

approved