OFFSET
1,1
COMMENTS
Every cyclic permutation of the digits is a prime, but there exists a non-cyclic permutation of the digits that produces a composite. [Extended by Felix Fröhlich, Aug 05 2018]
Conjecture: The sequence is finite, with 999331 being the last term (cf. A293142). - Felix Fröhlich, Aug 05 2018
LINKS
J. L. Boal and J. H. Bevis, Permutable primes, Mathematics Magazine, Vol. 55, No. 1 (1982), 38-41.
EXAMPLE
197, 719 and 971 are terms of the sequence, because all three numbers are prime, each number can be obtained by cyclically permuting the digits of one of the other numbers and there exist some composites, namely 791 and 917, that can be obtained from non-cyclic permutations of the digits of those three numbers. - Felix Fröhlich, Aug 10 2018
MATHEMATICA
Select[Prime@ Range@ PrimePi[10^6], Union[d = IntegerDigits[#], {1, 3, 7, 9}] == {1, 3, 7, 9} && AllTrue[ RotateLeft[d, #] & /@ Range@ IntegerLength@ #, PrimeQ@ FromDigits@ # &] && AnyTrue[ FromDigits /@ Permutations[d], CompositeQ] &] (* Giovanni Resta, Aug 10 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_circularprime(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))))
find_index_a(vec) = my(r=#vec-1); while(1, if(vec[r] < vec[r+1], return(r)); r--; if(r==0, return(-1)))
find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec
reverse_order(vec, r) = my(v=[], w=[]); for(x=1, r, v=concat(v, vec[x])); for(y=r+1, #vec, w=concat(w, vec[y])); w=Vecrev(w); concat(v, w)
next_permutation(vec) = my(r=find_index_a(vec)); if(r==-1, return(0), my(s=find_index_b(r, vec)); vec=switch_elements(vec, r, s); vec=reverse_order(vec, r)); vec
is_permutable_prime(n) = if(n < 10, return(1)); my(d=vecsort(digits(n))); while(1, if(!ispseudoprime(eva(d)), return(0)); d=next_permutation(d); if(d==0, return(1)))
forprime(p=1, , if(is_circularprime(p) && !is_permutable_prime(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 05 2018
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
N. J. A. Sloane, Jan 19 2012
EXTENSIONS
More terms from Felix Fröhlich, Aug 05 2018
STATUS
approved