A038618 -id:A038618 - cp's OEIS frontend

A038617 Primes not containing the digit '9'.

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 83, 101, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 181, 211, 223, 227, 233, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 383, 401, 421

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A007095. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Indranil Ghosh, Table of n, a(n) for n = 1..50000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Intersection of A000040 (primes) and A007095 (numbers with no digit 9).
Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and this sequence, respectively.
Primes with other restrictions on digits: A106116, A156756.

[ p: p in PrimesUpTo(500) | not 9 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[1000]], DigitCount[ # ][[9]] == 0 &] (* Stefan Steinerberger, May 20 2006 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 9), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

lista(nn) = forprime (p=2, nn, if (vecmax(digits(p)) != 9, print1(p, ", "))); \\ Michel Marcus, Apr 06 2016

next_A038617(n)=until((n=nextprime(n+1))==(n=next_A007095(n-1)), ); n \\ M. F. Hasler, Jan 14 2020

from sympy import isprime
i = 1
while i <= 300:
    if "9" not in str(i) and isprime(i):
        print(str(i), end=",")
    i += 1 # Indranil Ghosh, Feb 07 2017

a(n) ~ n^(log 10/log 9) * log(n). - Charles R Greathouse IV, Aug 03 2023

A038604 Primes not containing the digit '2'.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052404. - Michel Marcus, Feb 21 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Indranil Ghosh, Table of n, a(n) for n = 1..50000
M. F. Hasler, Numbers avoiding certain digits OEIS wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Cf. A000040, A052404.
Subsequence of A065091 (odd primes).
Primes having no digit d = 0..9 are A038618, A038603, this sequence, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(400) | not 2 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 2] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn, d=2) = {forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), d), print1(p, ", ")););} \\ Michel Marcus, Feb 21 2015

select( {is_A038604(n)=is_A052404(n)&&isprime(n)}, [1..400]) \\ see Wiki for more
{next_A038604(n)=until((n==nextprime(n+1))==n=next_A052404(n-1),);n} \\ M. F. Hasler, Jan 12 2020

from sympy import isprime, nextprime
def is_A038604(n): return str(n).find('2')<0 and isprime(n)
def next_A038604(n): # get smallest term > n
    while True:
        n = nextprime(n); s = str(n); t = s.find('2')
        if t < 0: return n
        t = 10**(len(s)-1-t); n += t - n%t
def A038604_upto(stop=math.inf, start=3):
    while start < stop: yield start; start = next_A038604(start)
list(A038604_upto(400))
# M. F. Hasler, Jan 12 2020

Intersection of A000040 and A052404. - M. F. Hasler, Jan 11 2020

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A038612 Primes not containing the digit '4'.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052406. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Jason Bard, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits OEIS wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Intersection of A000040 (primes) and A052406 (numbers without digit 4).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, this sequence, A038613, A038614, A038615, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(300) | not 4 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 4] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 4), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

( {A038612_upto(N)=select( is_A052406, primes([1, N]))} )(444) \\ or better:
next_A038612(n)={until(isprime(n), n=next_A052406(nextprime(n+1)-1)); n}
( {A038612_vec(n,M=1)=M--;vector(n,i, n=next_A038612(if(i>1, n)))} )(20, 1000)
\\ (See the OEIS wiki page for more.) - M. F. Hasler, Jan 12 2020

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A038614 Primes not containing the digit '6'.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052414. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits OEIS wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Intersection of A000040 (primes) and A052414 (numbers with no digit 6).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, this sequence, A038615, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(400) | not 6 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

no6:= proc(n) option remember;
  n mod 10 <> 6 and procname(floor(n/10))
end proc:
no6(0):= true:
select(no6 and isprime, [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 16 2017

Select[Prime[Range[70]], DigitCount[#, 10, 6] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 6), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

select( {is_A038614(n)=is_A052414(n)&&isprime(n)}, [1..350]) \\ see A052414
(A038614_upto(n)=select( is_A038614, primes([1,n])))(350) \\ needs the above
next_A038614(n)={until(isprime(n), n=next_A052414(nextprime(n+1)-1));n}
(A038614_vec(n)=vector(n,i,n=next_A038614(if(i>1,n))))(66) \\ M. F. Hasler, Jan 12 2020

Intersection of A000040 and A052414. - M. F. Hasler, Jan 12 2020

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A038613 Primes not containing the digit '5'.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052413. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

			From _M. F. Hasler_, Jan 14 2020: (Start)
After a(85) = 499, the next prime without digit 5 is a(86) = 601.
After a(3734) = 49999, the next term is a(3735) = 60013.
After a(27273) = 499979, the next term is 600011.
After a(206276) = 4999999, the next term is 6000011. (End)

Jason Bard, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Intersection of A000040 (primes) and A052413 (numbers with no digit 5).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, this sequence, A038614, A038615, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(400) | not 5 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 5] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 5), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

(A038613_upto(n)=select( is_A052413, primes([1, n])))(350) \\ see A052413
next_A038613(n)={until(isprime(n), n=next_A052413(nextprime(n+1)-1)); n}
( {A038613_vec(n, M=1)=M--;vector(n, i, M=next_A038613(M))} )(20, 1000) \\ Compute n terms >= M. See also the OEIS wiki page. - M. F. Hasler, Jan 14 2020

a(n) ≍ n^(log 10/log 9) log n. - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A038616 Primes not containing the digit '8'.

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, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 293, 307

Offset: 1

Vasiliy Danilov (danilovv(AT)usa.net), Jul 15 1998

Subsequence of primes of A052421. - Michel Marcus, Feb 22 2015

Maynard proves that this sequence is infinite and in particular contains the expected number of elements up to x, on the order of x^(log 9/log 10)/log x. - Charles R Greathouse IV, Apr 08 2016

Indranil Ghosh, Table of n, a(n) for n = 1..50000
M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).
Index to entries for primes with digits in a given set

Intersection of A000040 (primes) and A052421 (numbers with no 8).

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, A038614, A038615, this sequence, and A038617, respectively.

[ p: p in PrimesUpTo(400) | not 8 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 8] == 0 &] (* Harvey P. Dale, Jan 24 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 8), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

next_A038616(n)=until((n=nextprime(n+1))==(n=next_A052421(n-1)), ); n \\ M. F. Hasler, Jan 14 2020

a(n) ~ n^(log 10/log 9) * log(n). - Charles R Greathouse IV, Aug 03 2023

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A056709 Naught-y primes, primes with noughts (or zeros).

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1201, 1301, 1303, 1307, 1409, 1601, 1607, 1609, 1709, 1801, 1901, 1907

Offset: 1

Robert G. Wilson v, Aug 10 2000

Intersection of A000040 and A011540. - Michel Marcus, Mar 12 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Chris Caldwell, Naughty prime, Prime Pages' Glossary (UTM). (Date?)

Cf. A000040 (primes), A011540 (numbers that contain a 0).
Complement, in A000040, of zeroless primes A038618.
Cf. A164968 (Naughty primes: most digits are 0).

[p:p in PrimesUpTo(2000)|0 in Intseq(p)]; // Marius A. Burtea, Jan 13 2020

Select[ Range[ 1, 2500, 2 ], PrimeQ[ # ] && Sort[ RealDigits[ # ][ [ 1 ] ] ][ [ 1 ] ] == 0 & ]
(* Second program: *)
Select[Prime@ Range@ 300, DigitCount[#, 10, 0] > 0 &] (* Michael De Vlieger, Jan 28 2020 *)

is(n)=isprime(n)&&vecsort(eval(Vec(Str(n))),,8)[1]==0

select( {is_A056709(n)=!vecmin(digits(n))&&isprime(n)}, [1..2000]) \\ Defines the characteristic function is_A; as check & example: select terms in [1..2000].
next_A056709(n)={until(!vecmin(digits(n)), n=nextprime(n+1));n} \\ Successor function: find smallest a(k) > n. Useful to get a vector of consecutive terms:
A056709_vec(n,M=99)=M--;vector(n,i,M=next_A056709(M)) \\ get n terms >= M (if given, else start with a(1)).  \\ M. F. Hasler, Jan 12 2020

from sympy import primerange
def aupto(lim): return [p for p in primerange(1, lim+1) if '0' in str(p)]
print(aupto(1910)) # Michael S. Branicky, Mar 11 2022

a(n) ~ n log n: almost all primes are in this sequence. - Charles R Greathouse IV, Jul 24 2012

A034302 Zeroless primes that remain prime if any digit is deleted.

Original entry on oeis.org

23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1499, 1997, 2239, 2293, 3137, 4919, 6173, 7433, 9677, 19973, 23833, 26833, 47933, 73331, 74177, 91733, 93491, 94397, 111731, 166931, 333911, 355933, 477797, 477977

Offset: 1

David W. Wilson

Chai Wah Wu, Table of n, a(n) for n = 1..118 (terms 1..79 from T. D. Noe, terms 80..103 from Charles R Greathouse IV)
StackExchange, Deleting any digit yields a prime

Cf. A034303, A034304, A034305, A051362, A010051, A038618.

import Data.List (inits, tails)
a034302 n = a034302_list !! (n-1)
a034302_list = filter f $ drop 4 a038618_list where
   f x = all (== 1) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011

rpnzQ[n_]:=Module[{idn=IntegerDigits[n]},Count[idn,0]==0 && And@@ PrimeQ[FromDigits/@ Subsets[IntegerDigits[n], {Length[idn]-1}]]]; Select[Prime[Range[40000]],rpnzQ]  (* Harvey P. Dale, Mar 24 2011 *)

is(n)=my(d=digits(n),t=2^#d-1); if(vecmin(d)==0, return(0)); for(i=0,#d-1, if(!isprime(fromdigits(vecextract(d,t-2^i))), return(0))); isprime(n) \\ Charles R Greathouse IV, Jun 23 2017

from itertools import product
from sympy import isprime
A034302_list, m = [23, 37, 53, 73], 7
for l in range(1,m-1): # generate all terms less than 10^m
    for d in product('123456789',repeat=l):
        for e in product('1379',repeat=2):
            s = ''.join(d+e)
            if isprime(int(s)):
                for i in range(len(s)):
                    if not isprime(int(s[:i]+s[i+1:])):
                        break
                else:
                    A034302_list.append(int(s)) # Chai Wah Wu, Apr 05 2021

A178550 Primes with exactly one digit 1.

Original entry on oeis.org

13, 17, 19, 31, 41, 61, 71, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 199, 241, 251, 271, 281, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661, 691, 701, 719, 751, 761, 821, 881, 919, 941, 971, 991

Offset: 1

Lekraj Beedassy, May 29 2010

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A043493 and A000040.

Cf. A038618, A178551, A178552, A178553, A178554, A178555, A178556, A178557, A178558.

Select[Prime[Range[200]],Count[IntegerDigits[#],1]==1 &] (* Stefano Spezia, Aug 29 2025 *)

from sympy import isprime
print([i for i in range(1000) if str(i).count('1') == 1 and isprime(i)]) # Daniel Starodubtsev, Mar 29 2020

a(n) >> n^k where k = log(10)/log(9) = 1.04795.... - Charles R Greathouse IV, Jan 21 2025

A069488 Primes > 100 in which every substring of length 2 is also prime.

Original entry on oeis.org

113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, 971, 1117, 1171, 1319, 1373, 1973, 1979, 2311, 2371, 2971, 3119, 3137, 3719, 3797, 4111, 4373, 6113, 6131, 6173, 6197, 6719, 6737

Offset: 1

Amarnath Murthy, Mar 30 2002

Minimum number of digits is taken to be 3 as all two-digit primes would be trivial members.
From Robert G. Wilson v, May 12 2014: (Start)
The number of terms below 10^n: 0, 0, 21, 46, 123, 329, 810, 1733, 3985, 9710, ..., .
The least term with n digits is:  113, 1117, 11113, 111119, ..., see A090534.
The largest term with n digits is: 971, 9719, 97973, 979717, ..., see A242377.
The digits 2, 4, 5, 6 and 8 can only appear at the beginning of the prime and the digit 0 never appears. But the digits 1, 3, 7 and 9 can appear anywhere, yet only 1,1 can appear as a pair.
\10^n
d\  1&2   3    4    5     6     7     8      9     10 Total % @ 10^10
  \
1     0  19   34  146   648  1162  2678   8037  22740   39.188034
2     0   0    3    6    27    18    66    175    449    0.816186
3     0  14   19   63   326   712  1526   3855  11040   19.403018
4     0   3    2   13    54    92   143    384   1031    1.895550
5     0   0    0    9    17    24    45    176    426    0.763995
6     0   4    6    4    24    66   146    233    630    1.224834
7     0  14   20  100   436   907  1980   5442  15421   26.875285
8     0   0    3    6    24    25    37    176    388    0.721797
9     0   9   13   38   157   361   763   1790   5125    9.111301
Total 0  63  100  385  1713  3367  7384  20268  57250  100.00000
(End)

			3719 is a term as the three substrings of length 2, i.e., 37, 71 and 19, are all prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..10101 (first 1000 terms from Reinhard Zumkeller)

Cf. A069489 and A069490.

Cf. A010051, subsequence of zeroless primes: A038618.

a069488 n = a069488_list !! (n-1)
a069488_list = filter f $ dropWhile (<= 100) a038618_list where
   f x = x < 10 || a010051 (x `mod` 100) == 1 && f (x `div` 10)
-- Reinhard Zumkeller, Apr 07 2014

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 2, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[100] + 1, 500}]

Edited, corrected and extended by Robert G. Wilson v, Apr 12 2002

cp's OEIS Frontend

A038617 Primes not containing the digit '9'.

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A038604 Primes not containing the digit '2'.

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A038612 Primes not containing the digit '4'.

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A038614 Primes not containing the digit '6'.

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A038613 Primes not containing the digit '5'.

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A038616 Primes not containing the digit '8'.

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