A039986 -id:A039986 - cp's OEIS frontend

A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111

Offset: 1

N. J. A. Sloane

From Bill Gosper, Jan 24 2003, in a posting to the Math Fun Mailing List: (Start)
Recall Sloane's old request for more terms of A003459 = (2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 199 311 337 373 733 919 991 ...) and Richard C. Schroeppel's astonishing observation that the next term is 1111111111111111111. Absent Rich's analysis, trying to extend this sequence makes a great set of beginner's programming exercises. We may restrict the search to combinations of the four digits 1,3,7,9, only look at starting numbers with nondecreasing digits, generate only unique digit combinations, and only as needed. (We get the target sequence afterward by generating and merging the various permutations, and fudging the initial 2,3,5,7.)
To my amazement the (uncompiled, Macsyma) program printed 11,13,...,199,337, and after about a minute, 1111111111111111111!
And after a few more minutes, (10^23-1)/9! (End)
Boal and Bevis say that Johnson (1977) proves that if there is a term > 1000 with exactly two distinct digits then it must have more than nine billion digits. - N. J. A. Sloane, Jun 06 2015
Some authors require permutable or absolute primes to have at least two different digits. This produces the subsequence A129338. - M. F. Hasler, Mar 26 2008
See A039986 for a related problem with more sophisticated (PARI) code (iteration over only inequivalent digit permutations). - M. F. Hasler, Jul 10 2018

Richard C. Schroeppel, personal communication.
Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, pp. 20-21.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977. [Related paper, but primarily concerned with A023107 and A103443. - _N. J. A. Sloane_, Jun 06 2015]
T. N. Bhargava and P. H. Doyle, On the existence of absolute primes, Math. Mag., 47 (1974), 233.
J. L. Boal and J. H. Bevis, Permutable primes. Math. Mag., 55 (No. 1, 1982), 38-41.
C. Caldwell, The prime glossary: Permutable Prime.
J. P. Delahaye, Persistent Primes, Illustrating Permutable, Circular, Right & Left Truncatable Primes, Pour La Science no 256.
James Grime and Brady Haran, Absolute Primes, YouTube Numberphile video, 2024.
A. W. Johnson, Absolute primes, Mathematics Magazine, 1977, vol. 50, pp. 100-103.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
W. Schneider, MATHEWS, Circular, Permutable, Truncatable and Deletable Primes.
A. Slinko, Absolute Primes Oct. 2000.
A. Slinko, Absolute Primes, Oct. 2000 [Cached copy, permission requested].
Wikipedia, Permutable prime.
Index entries for sequences related to truncatable primes.

Includes all of A004022 = A002275(A004023).
A258706 gives minimal representatives of the permutation classes.
Cf. A010051, A023107, A016114, A103443, A129338, A141263.
Cf. A039986.

import Data.List (permutations)
a003459 n = a003459_list !! (n-1)
a003459_list = filter isAbsPrime a000040_list where
   isAbsPrime = all (== 1) . map (a010051 . read) . permutations . show
-- Reinhard Zumkeller, Sep 15 2011

f[n_]:=Module[{b=Permutations[IntegerDigits[n]],q=1},Do[If[!PrimeQ[c=FromDigits[b[[m]]]],q=0;Break[]],{m,Length[b]}];q];Select[Range[1000],f[#]>0&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2011 *)
(* Linear complexity: can't reach R(19). See A258706. - Bill Gosper, Jan 06 2017 *)

for(n=1, oo, my(S=[],r=10^n\9); for(a=1, 9^(n>1), for(b=if(n>2, 1-a), 9-a, for(j=0, if(b, n-1), ispseudoprime(a*r+b*10^j)||next(2)); S=concat(S,vector(if(b,n,1),k,a*r+10^(k-1)*b))));apply(t->printf(t","),Set(S))) \\ M. F. Hasler, Jun 26 2018

Conjecture: for n >= 23, a(n) = A004022(n-21). - Max Alekseyev, Oct 08 2018

The next terms are a(25)=A002275(317), a(26)=A002275(1031), a(27)=A002275(49081).

A096599 Squares k^2 with property that A062892(k^2) = 1.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 225, 289, 324, 361, 484, 529, 576, 676, 729, 784, 841, 1156, 1225, 1444, 1521, 1681, 1849, 2116, 2209, 2601, 2704, 3025, 3136, 3249, 3364, 3481, 3721, 3844, 3969, 4225, 4356, 4489, 4624, 5041, 5184, 5329, 5476

Offset: 1

Reinhard Zumkeller, Jun 29 2004

David A. Corneth, Table of n, a(n) for n = 1..6883

Cf. A096598, A096600, A000290, A000037, A039986.

from math import isqrt
from sympy.utilities.iterables import multiset_permutations as mp
def sqr(n): return isqrt(n)**2 == n
def ok(square):
    s = str(square)
    perms = (int("".join(p)) for p in mp(s, len(s)))
    return len(set(p for p in perms if sqr(p))) == 1
def aupto(limit): return [k*k for k in range(isqrt(limit)+1) if ok(k*k)]
print(aupto(5476)) # Michael S. Branicky, Oct 18 2021

Definition clarified by N. J. A. Sloane, Jan 16 2014

A374238 Primes whose pattern of identical digits is unique among the primes.

Original entry on oeis.org

11, 3333311, 7771717, 11818181, 515115551, 727722727, 887887787, 1110011101, 1161611161, 1411111441, 1411141411, 1717117117, 1911999919, 3311113111, 3313133311, 3333353533, 5151111551, 5555115151, 5777777557, 7373733337, 7747447777, 7777111777, 8887788787, 9199119991, 9994449499

Offset: 1

Dmytro Inosov, Jul 01 2024

The digit pattern for any natural number n is uniquely defined by the canonical form A358497(n), which enumerates digits according to their position of first occurrence. Each prime in this sequence has a unique digit pattern in the sense that no other prime has the same pattern.
Prime repunits (A004022) are a subsequence, as they are the sole primes with a single distinct digit.
A cryptarithm (alphametic) expresses a digit pattern in letters, where each distinct letter is to map to a distinct digit.
If a cryptarithmetic problem calls for a prime number, then the primes in this sequence are unique solutions, so we call these primes cryptarithmically unique.
The smallest term with 3 distinct digits is 1151135331533311.
The number of terms of length n is given by A376084(n).

			11 is a term since no other prime has the same pattern "AA" of two identical digits (any other AA is divisible by A > 1, hence nonprime).
Counterexample: 13 is not a term since another prime 17 has the same pattern "AB" of two nonidentical digits.
7771717 is a term since it's prime and no other prime has the same pattern "AAABABA".

Dmytro Inosov, Table of n, a(n) for n = 1..154
Dmytro Inosov, Table of n, a(n) for n = 1..24840
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024.
Wikipedia, Verbal arithmetic.

Cf. A000040 (primes), A004022 (prime repunits), A358497, A039986, A376918, A376084, A376118.

NumOfDigits = 10; (*Maximal integer length to be searched for*)
A358497[k_] :=
  FromDigits[
   Table[Mod[
     CountDistinct[Take[#, FirstPosition[#, #[[i]]][[1]]]] &[
      IntegerDigits[k]], 10], {i, 1, IntegerLength[k]}]];
A006880[MaxLen_] := PrimePi[10^MaxLen];
Extract[Select[
   Tally[Table[{#, A358497[#]} &[Prime[i]], {i, 1,
       A006880[NumOfDigits]}], #1[[2]] == #2[[2]] &], #[[2]] == 1 &], {All, 1}]

A166921 Least prime with exactly n prime anagrams not equal to itself.

Original entry on oeis.org

2, 13, 113, 149, 1013, 1039, 1427, 1123, 1439, 1579, 1237, 10271, 10453, 10139, 10253, 10243, 10457, 11579, 10789, 10273, 11239, 12457, 10729, 13249, 12347, 13687, 12539, 14759, 13799, 10739, 12637, 12893, 23957, 13597, 100493, 12379, 14593, 101383, 13789

Offset: 0

Pierre CAMI, Oct 23 2009

has only one prime anagram (31), and no smaller prime has a prime anagram other than itself, so a(1) = 13.
has 2 prime anagrams (131 and 311), and no smaller prime has two prime anagrams other than itself, so a(2) = 113.
has 3 prime anagrams (419, 491, and 941), and no smaller prime has three prime anagrams other than itself, so a(3) = 149.

			a(7) = prime 1123 with 7 prime anagrams 1213, 1231, 1321, 2113, 2131, 2311, 3121.

Michael S. Branicky, Table of n, a(n) for n = 0..2780 (terms 83..223 from P. CAMI and Chai Wah Wu, and terms 1..82 from P. CAMI)
Michael S. Branicky, Python program

Cf. A039986, A046810.

# see link for faster version
from sympy import isprime
from itertools import permutations
def anagrams(n):
  s = str(n)
  return set(int("".join(p)) for p in permutations(s) if p[0] != '0')
def num_prime_anagrams(n): return sum(isprime(i) for i in anagrams(n))
def a(n):
  if n == 0: return 2
  k = 3
  while not isprime(k) or num_prime_anagrams(k) != n+1: k += 2
  return k
print([a(n) for n in range(39)]) # Michael S. Branicky, Feb 13 2021

Definition edited and a(0) added by Chai Wah Wu, Dec 26 2016

A225421 Prime numbers consisting of only odd digits such that there is only one permutation of its digits that produces a prime number.

Original entry on oeis.org

3, 5, 7, 11, 19, 53, 59, 151, 353, 557, 599, 773, 997, 5557, 7559, 11119, 15559, 59999, 71777, 75553, 79999, 99991, 191999, 511111, 555557, 575557, 775777, 777977, 799979, 1111151, 3353333, 5595559, 5755559, 7577777, 9999991, 33335333, 55555553, 55555559

Offset: 1

T. D. Noe, May 07 2013

Cf. A039986 (similar, but allowing even digits also).

t2 = Select[Prime[Range[100000]], Intersection[{0, 2, 4, 6, 8}, Union[IntegerDigits[#]]] == {} &]; t = {}; Do[If[Length[Select[Table[FromDigits[k], {k, Permutations[IntegerDigits[p]]}], PrimeQ]] == 1, AppendTo[t, p]], {p, t2}]; t
edpQ[n_]:=Module[{idn=IntegerDigits[n]},AllTrue[idn,OddQ]&&Count[ FromDigits/@ Permutations[idn],?PrimeQ]==1]; Select[Prime[ Range[ 332*10^4]],edpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Nov 25 2018 *)

A364458 Prime numbers that are not repdigits with digits in nondecreasing order with the property that any nontrivial permutation of the digits gives a composite number.

Original entry on oeis.org

19, 23, 29, 47, 59, 67, 89, 223, 227, 229, 233, 257, 269, 449, 499, 557, 599, 677, 1447, 2267, 2447, 4447, 5557, 8999, 11119, 15559, 22229, 22669, 23333, 24889, 44449, 48889, 55589, 55889, 59999, 79999, 222269, 444449, 455557, 555557, 555589, 666667, 4444469, 4555559

Offset: 1

Jean-Marc Rebert, Dec 23 2023

The least terms with respectively 2, 3, 4 distinct digits are 19, 257, 24889.

			19 is a term, because the digits of 19 are in nondecreasing order and 91 is the unique number != 19 given by a permutation of 19 and 91 = 7 * 13 is composite and the digits of 91 are not in nondecreasing order.

Michael S. Branicky, Table of n, a(n) for n = 1..108 (all terms with <= 52 digits)
David A. Corneth, PARI program

Cf. A028864, A039986.

is(k)=my(u=digits(k),n=#u);if(#vecsort(u,,8)==1||u!=vecsort(u)||!isprime(k),return(0));forperm(n,p,my(vp=Vec(p),v=[]);for(i=1,n,v=concat(v,u[vp[i]]));q=fromdigits(v);if(k!=q&&isprime(q),return(0)));1

PARI
```
\\ See PARI link
```

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def bgen(d): yield from ("".join(m) for m in mc("123456789", d))
def agen(): yield from (t for d in count(1) for k in bgen(d) if len(set(k))!=1 and isprime(t:=int(k)) if not any((j:="".join(m))!=k and isprime(int(j)) for m in mp(k)))
print(list(islice(agen(), 44))) # Michael S. Branicky, Dec 23 2023

A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

Original entry on oeis.org

2, 2039, 6143, 522239, 33546239, 260046847, 16911433727, 32212254719, 2196875771903, 140735340871679, 2251799813685119, 9005000231485439, 576460752169205759, 36893488147410714623, 147573811852188057599, 9444732965739282038783, 154742504910672534362390399

Offset: 1

Ya-Ping Lu, Dec 23 2023

It seems that most of the terms end with '9', followed by those ending with '3', '7', and '1'.

			2 is a term because 2 is a prime with one '0' in binary form ('10') and '01' is not a prime. 2039 is a term because 2039 is a prime with one '0' in binary form ('11111110111') and changing the position of the '0', for example, '11111111011' = 2043 and '01111111111' = 1023, always results in a composite.

Subsequence of A095078.

Cf. A039986, A081118, A095058, A138290, A208083.

from sympy import isprime
for n in range(1,100):
    s = n*'1'; c = 0
    for j in range(n+1):
        num = int(s[:j]+'0'+s[j:], 2)
        if isprime(num):
            c += 1
            if c == 1: r = num
            if c == 2: break
    if c == 1: print(r, end = ', ')

cp's OEIS Frontend

A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A096599 Squares k^2 with property that A062892(k^2) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Extensions

A374238 Primes whose pattern of identical digits is unique among the primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A166921 Least prime with exactly n prime anagrams not equal to itself.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions

A225421 Prime numbers consisting of only odd digits such that there is only one permutation of its digits that produces a prime number.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A364458 Prime numbers that are not repdigits with digits in nondecreasing order with the property that any nontrivial permutation of the digits gives a composite number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python