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%I #24 Aug 26 2018 10:38:52
%S 197,719,971,1193,1931,3119,3779,7793,7937,9311,9377,11939,19391,
%T 19937,37199,39119,71993,91193,93719,93911,99371,193939,199933,319993,
%U 331999,391939,393919,919393,933199,939193,939391,993319,999331
%N Cyclic primes that are not absolute primes (A003459).
%C 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]
%C The sequence is the relative complement of A317688 in A293663. - _Felix Fröhlich_, Aug 05 2018
%C Conjecture: The sequence is finite, with 999331 being the last term (cf. A293142). - _Felix Fröhlich_, Aug 05 2018
%H J. L. Boal and J. H. Bevis, <a href="http://www.jstor.org/stable/2689862">Permutable primes</a>, Mathematics Magazine, Vol. 55, No. 1 (1982), 38-41.
%e 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
%t 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 *)
%o (PARI) eva(n) = subst(Pol(n), x, 10)
%o 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
%o 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))))
%o find_index_a(vec) = my(r=#vec-1); while(1, if(vec[r] < vec[r+1], return(r)); r--; if(r==0, return(-1)))
%o find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
%o switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec
%o 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)
%o 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
%o 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)))
%o forprime(p=1, , if(is_circularprime(p) && !is_permutable_prime(p), print1(p, ", "))) \\ _Felix Fröhlich_, Aug 05 2018
%Y Cf. A003459, A068652, A293142, A293663, A317688.
%K nonn,base,more
%O 1,1
%A _N. J. A. Sloane_, Jan 19 2012
%E More terms from _Felix Fröhlich_, Aug 05 2018
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