A103443 -id:A103443 - 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).

A023107 Largest integer in which every prefix is prime in base n (written in base 10).

Original entry on oeis.org

71, 191, 2437, 108863, 6841, 4497359, 1355840309, 73939133, 6774006887, 18704078369, 122311273757, 6525460043032393259, 927920056668659, 16778492037124607, 4928397730238375565449, 5228233855704101657, 3013357583408354653, 1437849529085279949589, 101721177350595997080671, 185720479816277907890970001

Offset: 3

David W. Wilson

Or, largest right-truncatable prime with base n>2 (in decimal form).

a(n) <= A094335(n).

			a(100) = 70123916363515199416199518301698321195339012727994799// 190371992151279729974757397909992327936943877127375781091143. - _Giovanni Resta_, Apr 11 2016

Martin Renner, Max Alekseyev and Giovanni Resta, Table of n, a(n) for n = 3..100
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
N. J. A. Sloane, The Angell-Godwin table of initial terms of A023107 and A103443
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A076586.

a(n) = my(S,m,D); D=select(x->(gcd(x,n)==1),vector(n-1,j,j)); S=select(ispseudoprime,vector(n-1,i,i)); while(#S, m=vecmax(S); S=concat(vector(#D,j,select(ispseudoprime,vector(#S,i,S[i]*n+D[j]))));); m /* Max Alekseyev, Dec 09 2014 */

More terms from Don Reble, Jun 23 2004
a(54)-a(75) in b-file from Max Alekseyev, Dec 09 2014
a(76)-a(100) in b-file from Giovanni Resta, May 01 2016

A076623 Total number of left truncatable primes (without zeros) in base n.

Original entry on oeis.org

0, 3, 16, 15, 454, 22, 446, 108, 4260, 75, 170053, 100, 34393, 9357, 27982, 362, 14979714, 685, 3062899, 59131, 1599447, 1372, 1052029701, 10484, 7028048, 98336, 69058060, 3926

Offset: 2

Martin Renner, Oct 22 2002, Nov 03 2002, Sep 24 2007, Feb 20 2008, Apr 20 2008

Approximation of a(b) by (PARI) code: l(b)=c=b*(b-1)/log(b)/eulerphi(b);\ return(floor((primepi(b)-omega(b))*exp(c)/c)); - Robert Gerbicz, Nov 02 2008

a(24) = 1052029701 based on strong BPSW pseudoprimes. Other terms up to a(29) use proved primes. - Martin Fuller, Nov 24 2008

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Michael S. Branicky, String-based Python Program
Martin Fuller, Table of n, a(n) for n= 2..53, with question marks where unknown
Hans Havermann, A076623 Decomposed
Index entries for sequences related to truncatable primes

Cf. A024779, A024780, A024781, A024782, A024783, A024784, A024785, A076586, A103443, A103463.

Lton := proc(L,b) add( op(i,L)*b^(i-1),i=1..nops(L)) ; end proc:
A076623rec := proc(L,b) local a,d,Lext,p ; a := 0 ; for d from 1 to b-1 do Lext := [op(L),d] ; p := Lton(Lext,b) ; if isprime(p) then a := a+1 ;  a := a+procname(Lext,b) ; end if; end do: a ;end proc:
A076623 := proc(b) A076623rec([],b) ; end proc:
for b from 2 do print(b,A076623(b)) ; end do: # R. J. Mathar, Jun 01 2011

f(b)=ct=0;A=[0];n=-1;L=1;while(L,n++;B=vector(L*b);M=0;\
for(i=1,L,for(j=1,b-1,x=A[i]+j*b^n;if(isprime[x],M++;B[M]=x;ct++)));\
L=M;A=vector(L,i,B[i]));return(ct) \\ Robert Gerbicz, Oct 31 2008

# works for all n; link has faster string-based version for n < 37
from sympy import isprime, primerange
from sympy.ntheory.digits import digits
def fromdigits(digs, base):
    return sum(d*base**i for i, d in enumerate(digs))
def a(n):
    prime_lists, an = [(p,) for p in primerange(1, n)], 0
    while len(prime_lists) > 0:
        an += len(prime_lists)
        candidates = set(p+(d,) for p in prime_lists for d in range(1, n))
        prime_lists = [c for c in candidates if isprime(fromdigits(c, n))]
    return an
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Apr 27 2022

a(12) corrected from 170051 to 170053 by Martin Fuller, Oct 31 2008
a(18) corrected by Robert Gerbicz, Nov 02 2008
a(24)-a(29) from Martin Fuller, Nov 24 2008
Entries in a-file corrected by N. J. A. Sloane, Jun 02 2011

A254755 Left-truncatable composites: every decimal suffix is a composite number.

Original entry on oeis.org

4, 6, 8, 9, 14, 16, 18, 24, 26, 28, 34, 36, 38, 39, 44, 46, 48, 49, 54, 56, 58, 64, 66, 68, 69, 74, 76, 78, 84, 86, 88, 94, 96, 98, 99, 104, 106, 108, 114, 116, 118, 124, 126, 128, 134, 136, 138, 144, 146, 148, 154, 156, 158, 164, 166, 168

Offset: 1

Stanislav Sykora, Feb 15 2015

			549 is a member because 549, 49, and 9 are all composites.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A103443 (left-truncatable primes), A202260 (right-truncatable composites), A254750.

isComposite(n) = (n>2)&&(!isprime(n));
isLeftTruncatableComposite(n,b=10) = {my(k=b);if(!isComposite(n),return(0););while(n\k>0,if(!isComposite(n%k),return(0););k*=b);return(1);}

A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

23, 11, 67, 839, 37, 1867, 173, 739397, 79, 105691, 379, 37573, 647, 3389, 631, 202715129, 211, 155863, 1283, 787817, 439, 109893629, 577, 4195880189, 1811, 14474071, 379, 21335388527, 2203, 1043557, 2939, 42741029, 2767, 50764713107, 853, 65467229, 4409, 8524002457

Offset: 3

Felix Fröhlich, Jan 05 2019

			For n = 12: 105691 is 511B7 in base 12. Successively removing the leftmost digit yields the base-12 numbers 11B7, 1B7, B7 and 7. When converted to base 10, these are 2011, 283, 139 and 7, respectively, all primes. Successively removing the rightmost digit yields the base-12 numbers 511B, 511, 51 and 5. When converted to base 10, these are 8807, 733, 61 and 5, respectively, all primes. Since no larger prime with this property in base 12 exists (as proven by Daniel Suteu), a(12) = 105691.

Daniel Suteu, Table of n, a(n) for n = 3..64
Wikipedia, Truncatable prime

Cf. A020994, A023107, A076586, A076623, A103443, A323390, A323396.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
a(n) = vecmax(bothTruncatablePrimesInBase(n)); \\ for n>=3; Daniel Suteu, Jan 22 2019

a(n) <= min(A023107(n), A103443(n)). - Daniel Suteu, Feb 24 2019

a(17)-a(40) from Daniel Suteu, Jan 11 2019

A103463 Length of the largest left-truncatable prime (in base n).

Original entry on oeis.org

0, 3, 6, 6, 17, 7, 15, 10, 24, 9, 32, 8, 26, 22, 25, 11, 43, 14, 37, 27, 37, 17, 53, 20, 39, 28, 46, 19

Offset: 2

Martin Renner, Mar 21 2005, Feb 20 2008, Apr 20 2008

The next term (base 30) will be difficult to calculate because there are over a trillion left-truncatable primes in that base for each of digit-lengths 29-34. Nevertheless, the largest left-truncatable prime in this base can be estimated by theory to have a length of about 82. [Hans Havermann, Aug 16 2011]

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Roman Maeder, A Prime Pencil
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A076623, A103443.

a(24)-a(29) from Hans Havermann, Aug 16 2011

A326609 Largest minimal prime in base n (written in base 10).

Original entry on oeis.org

3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921

Offset: 2

Richard N. Smith, Jul 13 2019

a(13) is (probably) 13^32020*8+183, it has 35670 digits, a(14) = 14^85*4+65, it has 99 digits, a(15) = (15^106*66-619)/7, it has 126 digits, a(16) = 16^3544*9+145, it has 4269 digits.
a(17) is the smallest prime of the form (4105*17^k-9)/16 if it exists, otherwise (probably) (73*17^111333-9)/16 (136991 digits), a(18) = 18^31*304+1 (42 digits).
Other known terms: a(20) = (20^449*16-2809)/19 (585 digits), a(22) = 22^763*20+7041 (1026 digits), a(23) is (probably) (23^800873*106-7)/11 (1090573 digits), a(24) = (24^99*512-121)/23 (138 digits), a(30) = 30^1023*12+1 (1513 digits), a(42) = (42^487*27-1093)/41 (791 digits).
a(19) is the smallest prime of the form (15964*19^k-1)/3 if it exists, otherwise (probably) (904*19^110984-1)/3 (141924 digits), a(21) is the smallest prime of the form 16*21^k+335 if it exists, otherwise (probably) (51*21^479149-1243)/4 (633542 digits).

Richard N. Smith, Table of n, a(n) for n = 2..16
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, preprint, 2014.
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics, Volume 25, 2016 - Issue 3.
C. K. Caldwell, The Prime Glossary, minimal prime
Richard N. Smith, List of all known terms
Top PRPs, Search for 8*13^n+183
Top PRPs, Search for (73*17^n-9)/16
Top PRPs, Search for (106*23^n-7)/11
Wikipedia, Minimal prime (recreational mathematics)

Cf. A071062 (base 10 minimal primes), A110600 (base 12 minimal primes).

Cf. A293142 (largest non-repunit permutable prime), A317689 (largest non-repunit circular prime), A103443 (largest left-truncatable prime), A023107 (largest right-truncatable prime), A323137 (largest two-sided prime), A084738 (smallest repunit prime), A186995 (smallest weakly prime).

cp's OEIS Frontend

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

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A023107 Largest integer in which every prefix is prime in base n (written in base 10).

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A076623 Total number of left truncatable primes (without zeros) in base n.

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A254755 Left-truncatable composites: every decimal suffix is a composite number.

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PARI

A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

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PARI

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A103463 Length of the largest left-truncatable prime (in base n).

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A326609 Largest minimal prime in base n (written in base 10).

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