cp's OEIS Frontend

A317688 Absolute primes that are not repunits: primes where the number resulting from any permutation of the digits is also prime, excluding repunit primes.

%I A317688 #17 Nov 22 2018 02:42:08
%S A317688 2,3,5,7,13,17,31,37,71,73,79,97,113,131,199,311,337,373,733,919,991
%N A317688 Absolute primes that are not repunits: primes where the number resulting from any permutation of the digits is also prime, excluding repunit primes.
%C A317688 Any term with two or more digits contains exactly two different digits from the set {1, 3, 7, 9} (cf. Erdős et al., 1977, Solution 953).
%C A317688 Conjecture: The sequence is finite, with 991 being the last term.
%C A317688 The known terms are those terms of A293663 where membership in A293663 trivially implies membership in this sequence, i.e., the numbers resulting from all cyclic permutations of the digits of these terms are the same as the numbers resulting from all permutations of the digits of these terms. This is the case only for terms with less than four digits.
%H A317688 P. Erdős et al., <a href="https://doi.org/10.2307/2689738">Problems</a>, Mathematics Magazine, Vol. 50, No. 2 (1977), 99-104.
%H A317688 Arkadii Slinko, <a href="https://arxiv.org/abs/1811.08613">Absolute Primes</a>, arXiv:1811.08613 [math.NT], 2018.
%H A317688 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutable_prime">Permutable prime</a>
%e A317688 The other numbers resulting from all possible permutations of the digits of 113 are 131 and 311. 113, 131 and 311 are all primes, so all three numbers are terms of this sequence.
%o A317688 (PARI) eva(n) = subst(Pol(n), x, 10)
%o A317688 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 A317688 find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
%o A317688 switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec
%o A317688 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 A317688 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 A317688 is(n) = if(n < 10, return(1)); my(d=vecsort(digits(n))); if(vecmin(d)==0 || vecmax(d)==1, return(0)); while(1, if(!ispseudoprime(eva(d)), return(0)); d=next_permutation(d); if(d==0, return(1)))
%o A317688 forprime(p=1, , if(is(p), print1(p, ", ")))
%Y A317688 Relative complement of A004022 in A003459. Supersequence of A129338. Subsequence of A293663.
%K A317688 nonn,base,more
%O A317688 1,1
%A A317688 _Felix Fröhlich_, Aug 04 2018