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A039986
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Primes such that every distinct permutation of digits is composite (including permutations with leading zeros).
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7
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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; i<n|| bittest(650, d[t])|| return; return(p=concat([setminus(Set(p[1..i]), [t]), t, p[i+1..n]]))), L=List(), f, p, d); forvec(i=vector(n, i, [7^(i==n), 10]), vecsum(d=vecextract(D, i))%3|| next; f=0; p=[1..n]; until(!nextperm(), isprime(vecextract(d, p)*u)&& (f&& next(2)|| f=p)); f&& d[f[1]]&& listput(L, vecextract(d, f)*u), 1); Set(L), primes(4))} \\ Returns all terms with n digits. - M. 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)
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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