A023107 -id:A023107 - cp's OEIS frontend

A024770 Right-truncatable primes: every prefix is prime.

2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193

Offset: 1

David W. Wilson

Primes in which repeatedly deleting the least significant digit gives a prime at every step until a single-digit prime remains. The sequence ends at a(83) = 73939133 = A023107(10).
The subsequence which consists of the following "chain" of consecutive right truncatable primes: 73939133, 7393913, 739391, 73939, 7393, 739, 73, 7 yields the largest sum, compared with other chains formed from subsets of this sequence: 73939133 + 7393913 + 739391 + 73939 + 7393 + 739 + 73 + 7 = 82154588. - Alexander R. Povolotsky, Jan 22 2008
Can also be seen as a table whose n-th row lists the n-digit terms; row lengths (0 for n >= 9) are given by A050986. The sequence can be constructed starting with the single-digit primes and appending, for each p in the list, the primes within 10*p and 10(p+1), formed by appending a digit to p. - M. F. Hasler, Nov 07 2018

Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer London 2010, pp. 86-89.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 112-113.

Jens Kruse Andersen, Table of n, a(n) for n = 1..83 (The full list of terms, taken from link below)
Jens Kruse Andersen, Right-truncatable primes
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Patrick De Geest, The list of 4260 left-truncatable primes
R. Schroeppel, HAKMEM item 33; "Russian Doll Primes", but with a slightly different definition.
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Supersequence of A085823, A202263. Subsequence of A012883, A068669. - Jaroslav Krizek, Jan 28 2012
Supersequence of A239747.
Cf. A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866, A077390 (left-and-right-truncatable primes), A137812 (left-or-right truncatable primes), A254751, A254753.
Cf. A237600 for the base-16 analog.

import Data.List (inits)
a024770 n = a024770_list !! (n-1)
a024770_list = filter (\x ->
   all (== 1) $ map (a010051 . read) $ tail $ inits $ show x) a038618_list
-- Reinhard Zumkeller, Nov 01 2011

s:=[1,3,7,9]: a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: do for j from l1 to l2 do for k from 1 to 4 do d:=[s[k],op(a[j])]: if(isprime(op(convert(d, base, 10, 10^nops(d)))))then a:=[op(a), d]: fi: od: od: l1:=l2+1: l2:=nops(a): if(l1>l2)then break: fi: od: seq(op(convert(a[j], base, 10, 10^nops(a[j]))),j=1..nops(a)); # Nathaniel Johnston, Jun 21 2011

max = 100000; truncate[p_] := If[PrimeQ[q = Quotient[p, 10]], q, p]; ok[p_] := FixedPoint[ truncate, p] < 10; p = 1; A024770 = {}; While[ (p = NextPrime[p]) < max, If[ok[p], AppendTo[ A024770, p]]]; A024770 (* Jean-François Alcover, Nov 09 2011, after Pari *)
eppQ[n_]:=AllTrue[FromDigits/@Table[Take[IntegerDigits[n],i],{i, IntegerLength[ n]-1}], PrimeQ]; Select[Prime[Range[3400]],eppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 14 2015 *)

{fileO="b024770.txt";v=vector(100);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4;j1=1; write(fileO,"1 2");write(fileO,"2 3");write(fileO,"3 5");write(fileO,"4 7"); until(0,if(j1>j,break);new=1;for(i=j1,j,if(new,j1=j+1;new=0);for(k=1,9, z=10*v[i]+k;if(isprime(z),j++;v[j]=z;write(fileO,j," ",z);))));} \\ Harry J. Smith, Sep 20 2008

for(n=2, 31193, v=n; while(isprime(n), c=n; n=(c-lift(Mod(c, 10)))/10); if(n==0, print1(v, ", ")); n=v); \\ Arkadiusz Wesolowski, Mar 20 2014

A024770=vector(9, n, p=concat(apply(t->primes([t, t+1]*10), if(n>1, p)))) \\ The list of n-digit terms, 1 <= n <= 9. Use concat(%) to "flatten" it. - M. F. Hasler, Nov 07 2018

from sympy import primerange
p = lambda x: list(primerange(x, x+10)); A024770 = p(0); i=0
while iA024770): A024770+=p(A024770[i]*10); i+=1 # M. F. Hasler, Mar 11 2020

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

Original entry on oeis.org

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).

A237600 Right-truncatable primes in base 16.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 37, 41, 43, 47, 53, 59, 61, 83, 89, 113, 127, 179, 181, 191, 211, 223, 593, 599, 601, 607, 659, 661, 691, 701, 757, 761, 853, 857, 859, 863, 947, 953, 977, 983, 991, 1427, 1429, 1433, 1439, 1811, 1823, 2039, 2879, 2897, 2903, 2909, 3061

Offset: 1

Stanislav Sykora, Feb 15 2014

Numbers with these properties: (i) a(n) is a prime and (ii) its image under the function E(k) = k\16 = floor(k/16) is zero or has the same properties. [Corrected by M. F. Hasler, Nov 07 2018]
The sequence has 414 nonzero members.
Otherwise said, integers p > 0 such that floor(p/16^k) is prime or zero for all k >= 0. One might relax to p >= 0, i.e., include an initial term 0, corresponding to an empty string of digits. The recursive definition can also be used to produce all of the terms, starting with the primes < 16, and adding, for each term of the list, the primes made from appending a digit to that term, i.e., the primes between 16 x that term and 16 more. The sequence can also be seen as a table whose n-th row yields the terms with n digits in base 16: row lengths are A237601 and the last term of row n is A237602(n). - M. F. Hasler, Nov 07 2018

			a(414) = 16778492037124607, in hexadecimal notation 3B9BF319BD51FF, belongs to a(n) because each of its hexadecimal prefixes (including itself) is a prime. Being the largest of such numbers, it is also a member of A023107.

Stanislav Sykora, Table of n, a(n) for n = 1..414
Stanislav Sykora, PARI/GP scripts for genetic threads

Cf. A023107, A024770 (base 10), A237601, A237602, A254756.

Select[Range@ 3600, AllTrue[Most[DeleteDuplicates@ FixedPointList[f, #]], PrimeQ] &] (* Michael De Vlieger, Mar 07 2015, Version 10 *)

GT_Trunc1(nmax,prop,b=10) = { \\ See the link for details
  my (n=0,v=vector(nmax),g=1,lgs=1,lge,an,c);
  for (k=1,b-1,if (prop(k),v[n++]=k));
  lge=n; c=lge-lgs+1;
  while (c, g++;for (k=lgs,lge,for (m=0,b-1, an=b*v[k]+m;
    if (prop(an), v[n++]=an;if (n>=nmax,return (v)));););
    lgs=lge+1; lge=n; c=lge-lgs+1;);
  if (n, return (v[1..n]));
  print("No solution");}
v = GT_Trunc1(1000000,isprime,16)

isok(n)={ while(n, if(!isprime(n),return(0));n\=16); 1} \\ Joerg Arndt, Mar 07 2015

my(A=primes([0,15]),i=1); until(#AA237600=A \\ M. F. Hasler, Nov 07 2018

from gmpy2 import is_prime
A237600_list = []
for n in range(1,10**9):
    if is_prime(n):
        s = format(n,'x')
        for i in range(1,len(s)):
            if not is_prime(int(s[:-i],16)):
                break
        else:
            A237600_list.append(n) # Chai Wah Wu, Apr 16 2015

from sympy import primerange
p = lambda x: list(primerange(x,x+16)); A237600 = p(0); i=0
while iA237600): A237600+=p(A237600[i]*16); i+=1 # M. F. Hasler, Mar 11 2020

A238853 Right-truncatable, reversible primes in base 256.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 773, 809, 823

Offset: 1

Stanislav Sykora, Mar 06 2014

See A238850 for definitions, and A238854 for comments on general context.

In base 256, there are 35127 such numbers (see A238855), shown here in decimal format. Base 256 is of interest to programmers because its digits correspond to 8-bit bytes and are easily readable in hexadecimal.

			The largest such number is 143496996325262301365903209731563 which, written in hex format, with hyphens between bytes for better readability, is 07-13-2F-CD-51-E1-B1-11-EB-23-CD-B3-15-EB. Truncate on the right any number of bytes and the remaining prefix is still a prime, no matter whether the bytes are read from left to right, or vice versa!

Stanislav Sykora, Table of n, a(n) for n = 1..35127
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. All in base 10: A238850, 16: A238851, 100: A238852.

Cf. In base n: A238854 (largest), A238856 (maximum digits), A238857 (m-digits counts). Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

A103443 Largest left-truncatable prime in base n (decimal expansion).

Original entry on oeis.org

23, 4091, 7817, 4836525320399, 817337, 14005650767869, 1676456897, 357686312646216567629137, 2276005673, 13092430647736190817303130065827539, 812751503, 615419590422100474355767356763

Offset: 3

Martin Renner, Mar 21 2005, Sep 24 2007, Apr 20 2008

Martin Fuller, Table of n, a(n) for n = 3..29
I. O. Angell, and H. J. Godwin, On Truncatable Primes Math. Comput. 31, 265-267, 1977. (See also Sloane link below)
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018).
Martin Renner, Table of n, a(n) for n = 3..53 (with some question marks). Corrected and expanded by Hans Havermann, Jan 25 2014.
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. A076623, A023107, A103463, A254755.

a(n)=my(v=primes(primepi(n-1)),u,t,b=1,best); while(#v, best=vecmax(v); b*=n; u=List(); for(i=1,#v,for(k=1,n-1,if(isprime(t=v[i]+k*b), listput(u,t)))); v=Vec(u)); best \\ Charles R Greathouse IV, Feb 05 2013

Base-14 entry corrected by Hans Havermann, May 30 2011
Corresponding entry in a-file corrected by N. J. A. Sloane, Jun 02 2011
a-file corrected and expanded by Hans Havermann, Jan 25 2014

A238852 Right-truncatable, reversible primes in base 100.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 311, 313, 347, 349, 353, 359, 367, 701, 709, 719, 727, 733, 739, 751, 769, 773, 787, 1103, 1109, 1123, 1163, 1181, 1193, 1301, 1303, 1319, 1321, 1327, 1361, 1777

Offset: 1

Stanislav Sykora, Mar 06 2014

See A238850 for definitions, and A238854 for comments on general context.

In base 100, chosen as one of four examples, there are 1552 such numbers.

			The largest number of this type (using hyphens to separate the base 100 digits) is 19-07-93-27-17-37-99-47-19-11.Truncate any even number of decimal digits on its right, and the remaining prefix is still a base-100 reversible prime (e.g., 19079327 and 27930719 are both primes).

Stanislav Sykora, Table of n, a(n) for n = 1..1552
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. All in base 10: A238850, 16: A238851, 256: A238853.
Cf. In base n: A238854 (largest), A238855 (totals), A238856 (maximum digits), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

A238854 Largest right-truncatable, reversible prime in base n.

Original entry on oeis.org

23, 53, 449, 191, 1171, 30671, 5827, 3733, 901687, 10357, 834469, 3043427, 5430889, 4060019, 498061, 34763, 118248433, 62344463, 218555173, 4463351, 114607657, 7903613, 14523874693, 211675817, 32814697, 93375223, 162466979, 8052409793, 12006877873

Offset: 3

Stanislav Sykora, Mar 07 2014

See A238850, A238851, A238852, A238853 for the finite lists of such numbers in four bases selected as examples. A sequence conceptually similar to this one, but for right-truncatable (not reversible!) primes is A023107. The present, more restrictive, condition leads to smaller numbers which can be evaluated in reasonable time for much higher n values.

			a(4) = 53 because it is a prime which in base 4 reads 311_b4, its reverse 113_b4 (decimal 23) is also a prime, the same holds for all its base-4 prefixes (31_b4 and 3_b4), and it is the largest natural having these properties.

Stanislav Sykora, Table of n, a(n) for n = 3..390
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
Cf. In base n: A238855 (totals), A238856 (maximum digits), A238857 (m-digits counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

A238855 Number of all right-truncatable reversible primes in base n.

Original entry on oeis.org

0, 3, 4, 12, 5, 12, 24, 17, 16, 33, 22, 29, 50, 39, 40, 39, 24, 65, 80, 100, 58, 58, 69, 122, 101, 90, 83, 125, 114, 133, 114, 122, 255, 203, 252, 123, 152, 221, 202, 308, 131, 250, 299, 397, 303, 143, 201, 484, 497, 423, 269, 253, 442, 944, 845, 378, 231, 460, 420, 455, 538, 438

Offset: 2

Stanislav Sykora, Mar 07 2014

For definitions and more comments, see A238854 and A238850.

Conjecture: in any base n, the number of right-truncatable reversible primes is finite.

			In bases 10, 16, 100, and 256 (used as examples in the crossrefs) there are, respectively, 16, 40, 1552, and 35127 such numbers.

Stanislav Sykora, Table of n, a(n) for n = 2..390
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments

Cf. Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
Cf. In base n: A238854 (largest), A238856 (maximum digits), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

A238856 Number of digits of the largest right-truncatable reversible prime in base n.

Original entry on oeis.org

0, 3, 3, 4, 3, 4, 5, 4, 4, 6, 4, 6, 6, 6, 6, 5, 4, 7, 6, 7, 5, 6, 5, 8, 6, 6, 6, 6, 7, 7, 6, 6, 8, 6, 8, 7, 8, 8, 7, 8, 7, 8, 8, 8, 8, 7, 7, 9, 9, 8, 7, 8, 10, 10, 9, 8, 6, 9, 8, 7, 9, 9, 9, 9, 11, 8, 7, 9, 10, 9, 10, 9, 9, 11, 10, 10, 9, 9, 8, 9, 9, 8, 10, 10, 10, 9, 9, 9, 10, 11

Offset: 2

Stanislav Sykora, Mar 13 2014

For definitions and more comments, see A238854 and A238850. A weak conjecture: this sequence might be bounded.

			a(16) = 6 because the largest truncatable reversible prime in base 16 has 6 hexadecimal digits (see A238851).

Stanislav Sykora, Table of n, a(n) for n = 2..390
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
Cf. In base n: A238854 (largest), A238855 (totals), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

A238857 Array read by rows: row n lists total number of m-digit right-truncatable reversible primes in base n.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 4, 4, 2, 0, 3, 1, 1, 0, 3, 5, 3, 1, 0, 4, 7, 7, 5, 1, 0, 4, 5, 5, 3, 0, 4, 5, 6, 1, 0, 4, 8, 7, 9, 4, 1, 0, 5, 5, 7, 5, 0, 5, 10, 8, 4, 1, 1, 0, 6, 11, 17, 12, 3, 1, 0, 6, 11, 13, 6, 2, 1, 0, 6, 9, 11, 9, 4, 1, 0, 6, 13, 12, 7, 1, 0, 7, 9, 7, 1, 0

Offset: 2

Stanislav Sykora, Mar 13 2014

For definitions and more comments, see A238854 and A238850.

This is an irregular table with one line for every base, starting at 2, while the columns correspond to the number of digits (1,2,3,...). Each row terminates with a zero (in any given base there appears to be a finite number of instances).

			These are the first rows of the table:
   2: 0,
   3: 1, 1, 1, 0,
   4: 2, 1, 1, 0,
   5: 2, 4, 4, 2, 0,
   6: 3, 1, 1, 0,
   7: 3, 5, 3, 1, 0,
   8: 4, 7, 7, 5, 1, 0,
   9: 4, 5, 5, 3, 0,
  10: 4, 5, 6, 1, 0,
  ...
Hence, there are 6 right truncatable reversible primes with 3 digits in base 10 (see A238850).

Stanislav Sykora, Table of n, a(n) for n = 2..5071
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
In base n: A238854 (largest), A238855 (totals), A238856 (maximum digits).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

cp's OEIS Frontend

A024770 Right-truncatable primes: every prefix is prime.

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

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A237600 Right-truncatable primes in base 16.

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A238853 Right-truncatable, reversible primes in base 256.

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A103443 Largest left-truncatable prime in base n (decimal expansion).

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A238852 Right-truncatable, reversible primes in base 100.

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A238854 Largest right-truncatable, reversible prime in base n.

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