A039986 Primes such that every distinct permutation of digits is composite (including permutations with leading zeros).

2, 3, 5, 7, 11, 19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 151, 211, 223, 227, 229, 233, 257, 263, 269, 353, 383, 409, 431, 433, 443, 449, 487, 499, 523, 541, 557, 599, 661, 677, 773, 827, 829, 853, 859, 881, 883, 887, 929, 997, 1447, 1451, 1481, 2111

Offset: 1

Author

David W. Wilson

Keywords

• base
• nonn

Comments

At most one permutation of digits of A179239 can occur in this sequence. - David A. Corneth, Jun 28 2018

Is there a term with more than 4 distinct digits? - David A. Corneth, Jun 30 2018

Up through 9999991 (the largest 7-digit prime) there are no terms with more than 4 distinct digits. - Harvey P. Dale, Dec 12 2018

The sequence can be seen as a table with the n-digit terms in row n. Row lengths would then be (4, 13, 34, 45, 68, 67, 47, 36, 40, 46, 33, 45, 35, 38, 32, ...). In these rows there are (0, 0, 0, 6, 9, 3, 0, 1, 0, 0, ...) terms with >= 4 distinct digits: this seems to happen only for terms with 4, 5, 6 or 8 digits. I conjecture that there are no more than these 6 + 9 + 3 + 1 = 19 terms (2861, 4027, 4801, 5209, 5623, 5849, 24889, 26561, 40609, 40883, 66541, 66853, 85087, 85843, 86441, 288689, 442469, 558541, 55555429) with 4, and none with 5 or more distinct digits. - M. F. Hasler, Jul 01 2018

Prime repunits (A004022) are a subset of this sequence. As larger terms are seemingly all near-repdigit primes, it is possible to obtain very large terms. For example: (10^10002 - 1)/9 - 10^2872. - Hans Havermann, Jul 08 2018

Links

• Hans Havermann and M. F. Hasler, Table of n, a(n) for n = 1..1141 (Terms < 10^30; earlier terms from T. D. Noe and David A. Corneth.)
• Hans Havermann, AEnlic primes.

Crossrefs

Cf. A030291, A179239.

Cf. A225421 (only odd digits).

Cf. A244529 for another variant. - M. F. Hasler, Jun 28 2018

Programs

Mathematica

t = {}; Do[p=Prime[n]; If[Length[Select[Table[FromDigits[k], {k,Permutations[IntegerDigits[p]]}], PrimeQ]] == 1, AppendTo[t,p]], {n,330}]; t (* Jayanta Basu, May 07 2013 *)
Select[Prime[Range[400]],AllTrue[FromDigits/@Rest[ Permutations[ IntegerDigits[#]]], CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 22 2015 *)

PARI

is(n,d=digits(n))={isprime(n)&&!for(i=1,(#d)!, (n=vecextract(d,numtoperm(#d,i)))!=d&& isprime(fromdigits(n))&& return)} \\ Then: select(is,primes(500)) - M. F. Hasler, Jun 28 2018
is(n)={isprime(n)||return; my(d=vecsort(digits(n), (a, b)->if(a-b&& t=bittest(650, a)-bittest(650, b),t,a-b)), p=vector(#d,i,i), N(p,i=2)= while((t=p[i]-1)&& while((setsearch(Set(p[i+1..#p]),t)|| d[t]==d[p[i]])&& t--,); !t, i++>#p&& return); i<#p|| bittest(650, d[t])|| return; concat([setminus(Set(p[1..i]),[t]), t, p[i+1..#p]]), t); #d==1|| !until(!p=N(p),(n!=t=fromdigits(vecextract(d,p)))&& isprime(t)&& return)} \\ Produces only inequivalent permutations which can be prime. - M. F. Hasler, Jun 28 2018
A039986_row(n)={if(n>1, local(D=eval(Vec("0245681379")), u=vectorv(n, i, 10^(n-i)), nextperm()=for(i=2,n,(t=p[i]-1)&& while(setsearch(Set(p[i+1..n]),t)|| d[t]==d[p[i]], t--||break); t|| next; iM. F. Hasler, Jul 01 2018

Python

from itertools import count, islice, combinations_with_replacement
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime
def A039986_gen(): # generator of terms
    for l in count(1):
        xlist = []
        for p in combinations_with_replacement('0123456789',l):
            flag = False
            for q in multiset_permutations(p):
                if isprime(m:=int(''.join(q))):
                    if flag or q[0]=='0':
                        flag = False
                        break
                    else:
                        flag = True
                        r = m
            if flag:
                xlist.append(r)
        yield from sorted(xlist)
A039986_list = list(islice(A039986_gen(),30)) # Chai Wah Wu, Dec 26 2023

Extensions

Name clarified upon the suggestion of Robert Israel, Jun 30 2018