A258706 -id:A258706 - 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).

A272106 Absolute primes in base 3: every permutation of digits in base 3 is a prime (only the smallest representatives of the permutation classes are shown).

Original entry on oeis.org

2, 5, 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013

Offset: 1

Chai Wah Wu, Apr 20 2016

For n <= 7, only a(2) = 5 is not a repunit in base 3. Supersequence of A076481. Base 3 analog of A258706.

Cf. A003459, A076481, A258706, A268812, A272107.

A258778 Least base b >= 2 such that prime(n) is an absolute prime in base b.

Original entry on oeis.org

3, 2, 3, 2, 5, 3, 5, 5, 4, 4, 2, 7, 7, 6, 7, 4, 8, 8, 9, 6, 8, 9, 11, 7, 7, 9, 11, 11, 13, 10, 2, 10, 12, 11, 13, 17, 12, 11, 12, 9, 16, 9, 6, 13, 15, 10, 6, 11, 19, 12, 19, 13, 11, 16, 7, 17, 19, 19, 12, 7, 16, 19, 7, 10, 13, 19, 22, 7, 19, 19, 18, 18, 21, 10

Offset: 1

Chai Wah Wu, Jun 11 2015

nonn

a(n) < prime(n) for n > 1. This is true since prime(n) in base prime(n)-1 is written as 11 which is an absolute prime.

Conjecture: a(n) < prime(n)-1 for n > 2.

			a(78) = 13. prime(78) = 397 in base 10 and 397_10 = 247_13. Rearranging the digits in base 13, we get 274_13 = 433_10, 427_13 = 709_10, 472_13 = 769_10, 724_13 = 1213_10, 742_13 = 1237_10, all of which are prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Wikipedia, Permutable prime

Cf. A003459, A258706, A258802.

A258802 Least base b >= 2 such that prime(n) is an absolute prime in base b with at least 2 distinct digits or 0 if no such base exists.

Original entry on oeis.org

0, 0, 3, 3, 5, 4, 5, 5, 4, 4, 6, 7, 7, 7, 7, 4, 8, 8, 9, 6, 9, 9, 11, 7, 7, 9, 11, 11, 13, 10, 13, 10, 12, 11, 13, 17, 14, 11, 12, 9, 16, 9, 6, 13, 15, 10, 6, 11, 19, 12, 19, 13, 11, 16, 7, 17, 19, 19, 12, 7, 16, 19, 7, 10, 13, 19, 22, 7, 19, 19, 18, 18, 21, 10

Offset: 1

Chai Wah Wu, Jun 11 2015

nonn

a(n) < prime(n)-1. This is true since prime(n) in base b > prime(n) has a single digit, prime(n) in base prime(n) is written as 10 which is not an absolute prime and prime(n) in base prime(n)-1 is written as 11 which does not have 2 distinct digits.
For 1 <= n <= 10000, A258778 differs from A258802 for 50 values of n.
Conjecture: a(n) = 0 if and only if n=1 or n=2.

			See example in A258778.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Wikipedia, Permutable prime

Cf. A003459, A258706, A258778.

A263499 Numbers with nondecreasing digits such that every cyclic shift is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, 3779, 1111111111111111111, 11111111111111111111111

Offset: 1

Chai Wah Wu, Nov 11 2015

a(16) is too big to display, see the b-file.

a(n) is the intersection of the sequences A068652 and A009994. A258706 is a subsequence. Up until a(16) only the term 3779 is missing from A258706.

Chai Wah Wu, Table of n, a(n) for n = 1..16

Cf. A068652, A009994, A258706.

fQ[n_] := Block[{d = IntegerDigits@ n}, And[d == Sort@ d, And @@ Table[PrimeQ@ FromDigits[d = RotateLeft@ d], {Length[d] - 1}]]]; Select[
Prime@ Range@ 600, fQ] (* Michael De Vlieger, Nov 12 2015, after T. D. Noe at A068652 *)

A268812 Absolute primes in base 16: every permutation of digits in base 16 is a prime (only the smallest representatives of the permutation classes are shown).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 53, 59, 61, 89, 191, 277, 283, 887, 1373, 1979, 3037

Offset: 1

Chai Wah Wu, Apr 20 2016

Base 16 analog of A258706.

Cf. A003459, A076481, A258706, A272106, A272107.

A272107 Absolute primes in base 8: every permutation of digits in base 8 is a prime (only the smallest representatives of the permutation classes are shown).

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 31, 47, 73, 1759

Offset: 1

Chai Wah Wu, Apr 20 2016

Base 8 analog of A258706.

Cf. A003459, A076481, A258706, A268812, A272106.

A317689 Largest nonrepunit base-n absolute prime (conjectured).

Original entry on oeis.org

7, 53, 3121, 211, 1999, 3803, 6469, 991, 161047, 19793, 16477, 24907, 683437, 3547, 67853, 80273, 94109, 72421

Offset: 3

Felix Fröhlich, Aug 04 2018

A base-b permutable or absolute prime is a prime p such that all numbers obtained from every permutation of the base-b digits of p and converted to base 10 are prime.

These primes were found using lim=10^8 in the PARI program and match those found with lim=10^5, lim=10^6 and lim=10^7. Therefore I conjecture that they are the correct values for those n.

Cf. A003459, A129338, A258706, A293142, 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)))
find_index_b(r, vec) = my(s=#vec); while(1, if(vec[r] < vec[s], return(s)); s--; if(s==r, return(-1)))
switch_elements(vec, firstpos, secondpos) = my(g); g=vec[secondpos]; vec[secondpos]=vec[firstpos]; vec[firstpos] = g; vec
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)
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
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_absolute_prime(n, base) = my(db=vecsort(digits(n, base))); if(vecmin(db)==0 || vecmax(db)==1, return(0)); while(1, my(dec=decimal(db, base)); if(!ispseudoprime(dec), return(0)); db=next_permutation(db); if(db==0, return(1)))
a(n) = my(absp=0, lim=10^7, i=0); forprime(p=n+1, , if(is_absolute_prime(p, n), absp=p); i++; if(i==lim, return(absp)))

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

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A272106 Absolute primes in base 3: every permutation of digits in base 3 is a prime (only the smallest representatives of the permutation classes are shown).

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A258778 Least base b >= 2 such that prime(n) is an absolute prime in base b.

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A258802 Least base b >= 2 such that prime(n) is an absolute prime in base b with at least 2 distinct digits or 0 if no such base exists.

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A263499 Numbers with nondecreasing digits such that every cyclic shift is a prime.

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Mathematica

A268812 Absolute primes in base 16: every permutation of digits in base 16 is a prime (only the smallest representatives of the permutation classes are shown).

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A272107 Absolute primes in base 8: every permutation of digits in base 8 is a prime (only the smallest representatives of the permutation classes are shown).

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A317689 Largest nonrepunit base-n absolute prime (conjectured).

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PARI