A327822 Numbers k such that when cyclically permuting the digits of k any number of times, any prime obtained is followed by a composite number and vice-versa.

14, 16, 19, 20, 23, 29, 30, 32, 34, 35, 38, 41, 43, 47, 50, 53, 59, 61, 67, 70, 74, 76, 83, 89, 91, 92, 95, 98, 1015, 1018, 1070, 1075, 1099, 1132, 1136, 1163, 1216, 1238, 1274, 1303, 1321, 1339, 1361, 1475, 1510, 1517, 1535, 1570, 1574, 1612, 1630, 1631, 1636

Offset: 1

Author

Felix Fröhlich, Sep 26 2019

Keywords

• nonn
• base

Examples

			When cyclically permuting the digits of 961990 one gets the numbers 961990, 619909, 199096, 990961, 909619, 96199 and these numbers are composite, prime, composite, prime, composite, prime, respectively, so 961990 (and each of these cyclic permutations except 96199) is a term of the sequence.
A more graphical representation:
       961990              C
      /      \           /   \
  096199   619909       P     P
     |        |         |     |
  909619   199096       C     C
      \      /           \   /
       990961              P
		

Crossrefs

Cf. A068652, A068654, A270083.

Programs

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(nn=#Str(n), u=[], v=vector(nn, x, x%2==0), w=vector(nn, x, x%2==1), d=digits(n), r=rot(d)); if(nn%2==1, return(0)); u=concat(u, [ispseudoprime(eva(d))]); u=concat(u, ispseudoprime(eva(r))); while(1, r=rot(r); if(r==d, if(u==v || u==w, return(1)); return(0)); u=concat(u, ispseudoprime(eva(r))))