A052023 -id:A052023 - 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

A024785 Left-truncatable primes: every suffix is prime and no digits are zero.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, 1223

Offset: 1

David W. Wilson

Last term is a(4260) = 357686312646216567629137 (Angell and Godwin). - Eric W. Weisstein, Dec 11 1999

Can be seen as table whose rows list n-digit terms, 1 <= n <= 25. Row lengths are A050987. - M. F. Hasler, Nov 07 2018

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

N. J. A. Sloane, Table of n, a(n) for n = 1..4260 (The full list, based on the De Geest web site)
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
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018).
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Rosetta Code, Programs for finding truncatable primes
Eric Weisstein's World of Mathematics, Truncatable Prime
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.
Index entries for sequences related to truncatable primes

Supersequence of A240768.

Cf. A033664, A032437, A020994, A024770 (right-truncatable primes), A052023, A052024, A052025, A050986, A050987, A077390 (left-and-right truncatable primes), A137812 (left-or-right truncatable primes), A254753.

import Data.List (tails)
a024785 n = a024785_list !! (n-1)
a024785_list = filter (\x ->
   all (== 1) $ map (a010051 . read) $ init $ tails $ show x) a038618_list
-- Reinhard Zumkeller, Nov 01 2011

a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: for n from 1 to 3 do for k from 1 to 9 do for j from l1 to l2 do d:=[op(a[j]),k]: 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
# second Maple program:
T:= proc(n) option remember; `if`(n=0, "", sort(select(isprime,
      map(x-> seq(parse(cat(j, x)), j=1..9), [T(n-1)])))[])
    end:
seq(T(n), n=1..4);  # Alois P. Heinz, Sep 01 2021

max = 2000; truncate[p_] := If[id = IntegerDigits[p]; FreeQ[id, 0] && (Last[id] == 3 || Last[id] == 7) && PrimeQ[q = FromDigits[ Rest[id]]], q, p]; ok[n_] := FixedPoint[ truncate, n] < 10;p = 5; A024785 = {2, 3, 5}; While[(p = NextPrime[p]) < max, If[ok[p], AppendTo[A024785, p]]]; A024785 (* Jean-François Alcover, Nov 09 2011 *)
d[n_]:=IntegerDigits[n]; ltrQ[n_]:=And@@PrimeQ[NestList[FromDigits[Drop[d[#],1]]&,n,Length[d[n]]-1]]; Select[Range[1225],ltrQ[#]&] (* Jayanta Basu, May 29 2013 *)
FullList=Sort[Flatten[Table[FixedPointList[Select[Flatten[Table[Range[9]*10^Length@IntegerDigits[#[[1]]] + #[[i]], {i, Length[#]}]], PrimeQ] &, {i}], {i, {2, 3, 5, 7}}]]] (* Fabrice Laussy, Nov 10 2019 *)

v=vector(4260);v[1]=2;v[2]=3;v[3]=5;v[4]=7;i=0;j=4; until(i>=j,i++;p=v[i];P10=10^(1+log(p)\log(10)); for(k=1,9,z=k*P10+p;if(isprime(z),j++;v[j]=z;))); s=vector(4260);s=vecsort(v);for(i=1,j,write("b024785.txt",i," ",s[i]);); \\

is_A024785(n,t=1)={until(t>10*p,isprime(p=n%t*=10)||return);n==p} \\ M. F. Hasler, Apr 17 2014

A024785=vector(25,n,p=vecsort(concat(apply(p->select(isprime, vector(9,i, i*10^(n-1)+p)),if(n>1,p))))); \\ Yields the list of rows (n-digit terms, n = 1..25). Use concat(%) to flatten. There are faster variants using matrices (vectorv(9,i,1)*p+[1..9]~*10^(n-1)*vector(#p,i,1)) and/or predefined vectors, but they are less concise and this takes less than 0.1 sec. - M. F. Hasler, Nov 07 2018

from sympy import isprime
def alst():
  primes, alst = [2, 3, 5, 7], []
  while len(primes) > 0:
    alst += sorted(primes)
    candidates = set(int(d+str(p)) for p in primes for d in "123456789")
    primes = [c for c in candidates if isprime(c)]
  return alst
print(alst()) # Michael S. Branicky, Apr 11 2021

A033664 Every suffix is prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 137, 167, 173, 197, 223, 283, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 503, 523, 547, 607, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 907, 937, 947

Offset: 1

N. J. A. Sloane

Primes in which repeatedly deleting the most significant digit gives a prime at every step until a single-digit prime remains.

Every digit string containing the least significant digit is prime. - Amarnath Murthy, Sep 24 2003

Alois P. Heinz, Table of n, a(n) for n = 1..66973 (first 8779 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Truncatable Prime.

Cf. A024785, A032437, A020994, A024770, A052023, A052024, A052025, A050986, A050987, A069866, A038680, A010051, A000040.

a033664 n = a033664_list !! (n-1)
a033664_list = filter (all ((== 1) . a010051. read) .
                           init . tail . tails . show) a000040_list
-- Reinhard Zumkeller, Jul 10 2013

T:= proc(n) option remember; `if`(n=0, "", select(isprime, [seq(seq(
      seq(parse(cat(j, 0$(n-i), p)), p=[T(i-1)]), i=1..n), j=1..9)])[])
    end:
seq(T(n), n=1..4);  # Alois P. Heinz, Sep 01 2021

h8pQ[n_]:=And@@PrimeQ/@Most[NestWhileList[FromDigits[Rest[ IntegerDigits[ #]]]&, n,#>0&]]; Select[Prime[Range[1000]],h8pQ] (* Harvey P. Dale, May 26 2011 *)

fileO="b033664.txt";lim=8779;v=vector(lim);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4; write(fileO,"1 2");write(fileO,"2 3");write(fileO,"3 5");write(fileO,"4 7"); p10=1;until(0,p10*=10;j0=j;for(k=1,9,k10=k*p10; for(i=1,j0,if(j==lim,break(3));z=k10+v[i]; if(isprime(z),j++;v[j]=z;write(fileO,j," ",z);)))) \\ Harry J. Smith, Sep 20 2008

from sympy import isprime, primerange
def ok(p): # does prime p satisfy the property
    s = str(p)
    return all(isprime(int(s[i:])) for i in range(1, len(s)))
print(list(filter(ok, primerange(1, 1000)))) # Michael S. Branicky, Sep 01 2021

# alternate for going to large numbers
def agen(maxdigits):
    yield from [2, 3, 5, 7]
    primestrs, digits, d = ["2", "3", "5", "7"], "0123456789", 1
    while len(primestrs) > 0 and d < maxdigits:
        cands = set(d+p for p in primestrs for d in "0123456789")
        primestrs = [c for c in cands if c[0] == "0" or isprime(int(c))]
        yield from sorted(map(int, (p for p in primestrs if p[0] != "0")))
        d += 1
print([p for p in agen(11)]) # Michael S. Branicky, Sep 01 2021

More terms from Erich Friedman

A020994 Primes that are both left-truncatable and right-truncatable.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

Two-sided primes: deleting any number of digits at left or at right, but not both, leaves a prime.
Primes in which every digit string containing the most significant digit or the least significant digit is prime. - Amarnath Murthy, Sep 24 2003
Intersection of A024785 and A024770. - Robert Israel, Mar 23 2015

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 178 (Rev. ed. 1997).

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
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A024770, A052023, A052024, A052025, A050986, A050987, A254751, A254753.

tspQ[n_] := Module[{idn=IntegerDigits[n], l}, l=Length[idn]; Union[PrimeQ/@(FromDigits/@ Join[Table[Take[idn, i], {i, l}], Table[Take[idn, -i], {i, l}]])]=={True}] Select[Prime[Range[PrimePi[740000]]], tspQ]

Corrected by David W. Wilson

Additional comments from Harvey P. Dale, Jul 10 2002

A050986 Number of n-digit right-truncatable primes.

Original entry on oeis.org

4, 9, 14, 16, 15, 12, 8, 5, 0

Offset: 1

Eric W. Weisstein

Right-truncatable means that the integer part of successive divisions by 10 always yields primes (or zero). - M. F. Hasler, Nov 07 2018

Jens Kruse Andersen, Right-truncatable primes
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes
Index entries for sequences related to numbers of primes in various ranges

Cf. A033664, A024785, A032437, A020994, A024770, A052023, A052024, A052025, A050987.

A050986=vector(9, n, #p=concat(apply(t->primes([t, t+1]*10), if(n>1, p)))) \\ M. F. Hasler, Nov 07 2018

Edited by Ray Chandler, Mar 13 2007

a(9) = 0 added by M. F. Hasler, Nov 07 2018

A050987 Number of n-digit left-truncatable primes.

Original entry on oeis.org

4, 11, 39, 99, 192, 326, 429, 521, 545, 517, 448, 354, 276, 212, 117, 72, 42, 24, 13, 6, 5, 4, 3, 1, 0

Offset: 1

Eric W. Weisstein

The sequence is well defined for any positive integer, with a(n) = 0 for n >= 25. But it makes sense to consider it to be full & finite. - M. F. Hasler, Nov 07 2018

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
P. De Geest, The list of 4260 left-truncatable primes
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to numbers of primes in various ranges
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A020994, A024770, A052023, A052024, A052025, A050986.

A050987=vector(25, n, #p=concat(apply(p->select(isprime, vector(9, i, i*10^(n-1)+p)), if(n>1, p)))) \\ M. F. Hasler, Nov 07 2018

from sympy import isprime
def alst():
  primes, alst = [2, 3, 5, 7], [4]
  while len(primes) > 0:
    candidates = set(int(d+str(p)) for p in primes for d in "123456789")
    primes = [c for c in candidates if isprime(c)]
    alst.append(len(primes))
  return alst
print(alst()) # Michael S. Branicky, Apr 11 2021

Edited by Ray Chandler, Mar 13 2007

a(25) = 0 added by M. F. Hasler, Nov 07 2018

A052025 Every prefix (or suffix) of palindromic prime a(n) is prime (right/left-truncatable).

Original entry on oeis.org

2, 3, 5, 7, 313, 373, 797

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. K. Caldwell, Left and Right truncatable primes.
P. De Geest, The list of 4260 left-truncatable primes
Eric Weisstein's World of Mathematics, Prime strings
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A020994, A024770, A052023, A052024, A050986, A050987.

A032437 Substrings from the right are prime numbers (using only odd digits different from 5).

Original entry on oeis.org

3, 7, 13, 17, 37, 73, 97, 113, 137, 173, 197, 313, 317, 337, 373, 397, 773, 797, 937, 997, 1373, 1997, 3137, 3313, 3373, 3797, 7937, 9137, 9173, 9337, 9397, 13313, 33797, 39397, 79337, 79397, 91373, 91997, 99137, 99173, 99397, 139397, 379397

Offset: 1

Carlos Rivera

Primes p with decimal expansion d_1 d_2 d_3 ... d_k such that the digits d_i are 1, 3, 7, or 9, and deleting 1, 2, 3, up to k-1 leading digits also produces a prime. For example, 9173 is a term because all of 9173, 173, 73, and 3 are primes. - N. J. A. Sloane, Jun 28 2022

			173 is a term because 173, 73, and 3 are all primes. 371 is not a term because 371 and 1 are not primes. - _N. J. A. Sloane_, Jun 28 2022

T. D. Noe, Table of n, a(n) for n = 1..58 [The complete list of terms]
C. Rivera, Prime strings
Eric Weisstein's World of Mathematics, Truncatable Prime.

Cf. A033664, A024785, A020994, A024770, A052023, A052024, A052025, A050986, A050987.

Select[Prime[Range[33000]],SubsetQ[{1,3,7,9},IntegerDigits[#]]&&AllTrue[Mod[#,10^Range[ IntegerLength[ #]-1]],PrimeQ]&] (* Harvey P. Dale, Jun 28 2022 *)

is(n)=my(d=digits(n)); for(i=1,n, if(!isprime(fromdigits(d[i..n])), return(0))); 1 \\ Charles R Greathouse IV, Jun 25 2017

Single-digit terms added by Eric W. Weisstein.

A052024 Every suffix of palindromic prime a(n) is prime (left-truncatable).

Original entry on oeis.org

2, 3, 5, 7, 313, 353, 373, 383, 797, 30103, 31013, 70607, 73037, 76367, 79397, 3002003, 7096907, 7693967, 700090007, 799636997, 70060906007, 3000002000003, 7030000000307, 300000020000003, 300001030100003, 310000060000013, 38000000000000000000083, 30000000004000300040000000003, 3000001000000000000000000000001000003

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. K. Caldwell, Left and Right truncatable primes.
Eric Weisstein's World of Mathematics, Prime strings
Index entries for sequences related to truncatable primes

Cf. A002385, A033664, A024785, A032437, A020994, A024770, A052023, A052025, A050986, A050987.

d[n_]:=IntegerDigits[n]; ltrQ[n_]:=And@@PrimeQ[NestWhileList[FromDigits[Drop[d[#],1]]&,n,#>9&]]; palQ[n_]:=Reverse[x=d[n]]==x; Select[Prime[Range[550000]],palQ[#]&<rQ[#]&] (* Jayanta Basu, Jun 02 2013 *)

from sympy import isprime
from itertools import count, islice
def agen(verbose=False):
    prime_strings, alst = {"3", "7"}, []
    yield from [2, 3, 5, 7]
    for digs in count(2):
        new_prime_strings = set()
        for p in prime_strings:
            for d in "123456789":
                ts = d + "0"*(digs-1-len(p)) + p
                if isprime(int(ts)):
                    new_prime_strings.add(ts)
        prime_strings |= new_prime_strings
        pals = [int(s) for s in new_prime_strings if s == s[::-1]]
        yield from sorted(pals)
        if verbose: print("...", digs, len(prime_strings), time()-time0)
print(list(islice(agen(), 20))) # Michael S. Branicky, Apr 04 2022

Inserted missing 31013 by Jayanta Basu, Jun 02 2013

a(27)-a(29) from Michael S. Branicky, Apr 04 2022

A173057 Partial sums of A024770.

Original entry on oeis.org

2, 5, 10, 17, 40, 69, 100, 137, 190, 249, 320, 393, 472, 705, 944, 1237, 1548, 1861, 2178, 2551, 2930, 3523, 4122, 4841, 5574, 6313, 7110, 9443, 11782, 14175, 16574, 19513, 22632, 25769, 29502, 33241, 37034, 40831, 46770, 53963, 61294, 68627

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of right-truncatable primes, primes whose every prefix is prime (in decimal representation). The sequence has 83 terms. The subsequence of prime partial sums of right-truncatable primes begins: 2, 5, 17, 137, 1237, 1861, 2551, 199483. What is the largest value in the subsubsequence of right-truncatable prime partial sums of right-truncatable primes?

			a(50) = 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 + 31379.

Cf. A000040, A024770, A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866.

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A024770 Right-truncatable primes: every prefix is prime.

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A024785 Left-truncatable primes: every suffix is prime and no digits are zero.

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A033664 Every suffix is prime.

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A020994 Primes that are both left-truncatable and right-truncatable.

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A050986 Number of n-digit right-truncatable primes.

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A050987 Number of n-digit left-truncatable primes.

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Python

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