A038603 -id:A038603 - cp's OEIS frontend

A375533 a(n) = numerator of Sum_{i=1..n} 1/A038603(i).

0, 1, 5, 31, 247, 5891, 175669, 6639823, 290694979, 13885515383, 746406329689, 44593096214321, 3020489689357037, 222690147603898211, 17752712881208877899, 1486130275559909484787, 133315968357656471537153, 13025132201814060676912631, 2913672358303309675918969343, 663425761972477930761347977351, 152383524508438692136746106396609

Offset: 0

N. J. A. Sloane, Sep 06 2024

Numerator of sum of reciprocals of primes with no digit "1".

Of interest because it appears that the value of Sum_{i=1..oo} 1/A038603(i) = lim_{i->oo} a(i)/A375534(i) is extremely difficult to compute - so difficult that its decimal expansion does not have an OEIS entry. (Compare A082830.)

			The first few sums are 0/1, 1/2, 5/6, 31/30, 247/210, 5891/4830, 175669/140070, 6639823/5182590,  ...

Alois P. Heinz, Table of n, a(n) for n = 0..313
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A024451, A038603, A375534, A082830.

a[n_]:=Numerator[Sum[ 1/Part[ResourceFunction["OEISSequence"]["A038603"],i],{i,n}]]; Array[a,20] (* Stefano Spezia, Sep 06 2024 *)

a(0) prepended by Alois P. Heinz, Oct 21 2024

A375534 a(n) = denominator of Sum_{i=1..n} 1/A038603(i).

Original entry on oeis.org

1, 2, 6, 30, 210, 4830, 140070, 5182590, 222851370, 10474014390, 555122762670, 32752242997530, 2194400280834510, 160191220500919230, 12655106419572619170, 1050373832824527391110, 93483271121382937808790, 9067877298774144967452630, 2022136637626634327741936490

Offset: 0

N. J. A. Sloane, Sep 06 2024

See A375533 (the numerators) for further information.

			The first few sums are 0/1, 1/2, 5/6, 31/30, 247/210, 5891/4830, 175669/140070, 6639823/5182590,  ...

Alois P. Heinz, Table of n, a(n) for n = 0..313
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A002110, A038603, A375533, A375805.

a[n_]:=Denominator[Sum[ 1/Part[ResourceFunction["OEISSequence"]["A038603"],i],{i,n}]]; Array[a,18] (* Stefano Spezia, Sep 06 2024 *)

a(0)=1 prepended by Alois P. Heinz, Oct 21 2024

A038618 Primes not containing the digit '0'.

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, 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, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

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

Complement of A056709 with respect to primes (A000040). - Lekraj Beedassy, Jul 04 2010

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
René-Louis Clerc, Nombres premiers primaires et nombres premiers secondaires , 2025.
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).
Eric Weisstein's World of Mathematics, Zerofree.
Index to entries for primes with digits in a given set

Cf. A010051, A168046, A384793, A384794.
Subsequence of A000040 (primes), A052382 (zeroless numbers) and A195943.
Primes having no digit d = 0..9 are this sequence, A038603, A038604, A038611, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

a038618 n = a038618_list !! (n-1)
a038618_list = filter ((== 1) . a168046) a000040_list
-- Reinhard Zumkeller, Apr 07 2014, Sep 27 2011

[ p: p in PrimesUpTo(300) | not 0 in Intseq(p) ];  // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 0] == 0 &] (* Vincenzo Librandi, Aug 09 2011 *)

is(n)=if(isprime(n),n=vecsort(eval(Vec(Str(n))),,8);n[1]>0) \\ Charles R Greathouse IV, Aug 09 2011

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

next_A038618(n)=until(vecmin(digits(n=nextprime(next_A052382(n)))),);n \\ Cf. OEIS Wiki page (LINKS) for other programs. - M. F. Hasler, Jan 12 2020

from sympy import primerange
def aupto(N): return [p for p in primerange(1, N+1) if '0' not in str(p)]
print(aupto(300)) # Michael S. Branicky, Mar 11 2022

Intersection of A052382 (zeroless numbers) and A000040 (primes); A168046(a(n))*A010051(a(n)) = 1. - Reinhard Zumkeller, Dec 01 2009

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

A038615 Primes not containing the digit '7'.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 23, 29, 31, 41, 43, 53, 59, 61, 83, 89, 101, 103, 109, 113, 131, 139, 149, 151, 163, 181, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 263, 269, 281, 283, 293, 311, 313, 331, 349, 353, 359, 383, 389, 401, 409, 419, 421, 431, 433, 439

Offset: 1

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

Subsequence of primes of A052419. - 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

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Marianne Freiberger, Primes without 7s, +plus magazine, August 2016.
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, A052419.

Primes having no digit d = 0..9 are A038618, A038603, A038604, A038611, A038612, A038613, A038614, this sequence, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(500) | not 7 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 7] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 7), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

(A038615_upto(N)=select( is_A052419, primes([1,N])))(444) \\ i.e.: {is_A038615(n)=is_A052419(n)&&isprime(n)}; {is_A052419(n)=!setsearch(Set(digits(n)),7)}. - M. F. Hasler, Jan 11 2020

Intersection of A000040 (primes) and A052419 (numbers with no digit 7). - 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

A052383 Numbers without 1 as a digit.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

For each k in {1, 2, ..., 29, 30, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43} there exists at least an m such that m^k is 1-less. If m^k is 1-less then (10*m)^k, (100*m)^k, (1000*m)^k, ... are also 1-less. Therefore for each of these numbers k there exist infinitely many k-th powers in this sequence. - Mohammed Yaseen, Apr 17 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS Wiki, Jan 12 2020.
Index entries for 10-automatic sequences.

Cf. A004176, A004720, A011531 (complement), A038603 (subset of primes), A082830 (Kempner series), A248518, A248519.

Cf. A052382 (without 0), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

a052383 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + if r > 0 then r + 1 else 0  where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 1 in Intseq(n) ];  // Bruno Berselli, May 28 2011

M:= 3: # to get all terms with up to M digits
B:= {$2..9}: A:= B union {0}:
for m from 1 to M do
B:= map(b -> seq(10*b+j,j={0,$2..9}), B);
A:= A union B;
od:
sort(convert(A,list)); # Robert Israel, Jan 11 2016
# second program:
A052383 := proc(n)
      option remember;
      if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if nops(convert(convert(a,base,10),set) intersect {1}) = 0 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 31 2016
# third Maple program:
a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d=0, 0, d+1))(irem(m, 9, 'm')), l
      od; parse(cat(l))/10
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 01 2016

ban1Q[n_] := FreeQ[IntegerDigits[n], 1] == True; Select[Range[0, 89], ban1Q[#] &] (* Jayanta Basu, May 17 2013 *)
Select[Range[0, 99], DigitCount[#, 10, 1] == 0 &] (* Alonso del Arte, Jan 12 2020 *)

a(n)=my(v=digits(n,9));for(i=1,#v,if(v[i],v[i]++));subst(Pol(v),'x,10) \\ Charles R Greathouse IV, Oct 04 2012

apply( {A052383(n)=fromdigits(apply(d->d+!!d, digits(n-1, 9)))}, [1..99]) \\ Defines the function and computes it for indices 1..99 (check & illustration)
select( {is_A052383(n)=!setsearch(Set(digits(n)), 1)}, [0..99]) \\ Define the characteristic function is_A; as illustration, select the terms in [0..99]
next_A052383(n, d=digits(n+=1))={for(i=1, #d, d[i]==1&& return((1+n\d=10^(#d-i))*d)); n} \\ Successor function: efficiently skip to the next a(k) > n. Used in A038603.  - M. F. Hasler, Jan 11 2020

from itertools import count, islice, product
def A052383(): # generator of terms
    yield 0
    for digits in count(1):
        for f in "23456789":
            for r in product("023456789", repeat=digits-1):
                yield int(f+"".join(r))
print(list(islice(A052383(), 72))) # Michael S. Branicky, Oct 15 2023

from gmpy2 import digits
def A052383(n): return int(''.join(str(int(d)+(d!='0')) for d in digits(n-1,9))) # Chai Wah Wu, Jun 28 2025

(0 to 99).filter(.toString.indexOf('1') == -1) // _Alonso del Arte, Jan 12 2020

seq 0 1000 | grep -v 1; # Joerg Arndt, May 29 2011

a(1) = 1, a(n + 1) = f(a(n) + 1, a(n) + 1) where f(x, y) = if x < 10 and x <> 1 then y else if x mod 10 = 1 then f(y + 1, y + 1) else f(floor(x/10), y). - Reinhard Zumkeller, Mar 02 2008
a(n) is the replacement of all nonzero digits d by d + 1 in the base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014
Sum_{k>1} 1/a(k) = A082830 = 16.176969... (Kempner series). - Bernard Schott, Jan 12 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A038611 Primes not containing the digit '3'.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 41, 47, 59, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 149, 151, 157, 167, 179, 181, 191, 197, 199, 211, 227, 229, 241, 251, 257, 269, 271, 277, 281, 401, 409, 419, 421, 449, 457, 461, 467, 479, 487, 491, 499, 509, 521, 541, 547, 557

Offset: 1

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

Subsequence of primes of A052405. - 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 A052405 (numbers with no digit 3).

Primes having no digit d = 0..9 are A038618, A038603, A038604, this sequence, A038612, A038613, A038614, A038615, A038616, and A038617, respectively.

[ p: p in PrimesUpTo(600) | not 3 in Intseq(p) ]; // Bruno Berselli, Aug 08 2011

Select[Prime[Range[70]], DigitCount[#, 10, 3] == 0 &] (* Vincenzo Librandi, Aug 08 2011 *)

lista(nn)=forprime(p=2, nn, if (!vecsearch(vecsort(digits(p),,8), 3), print1(p, ", "));); \\ Michel Marcus, Feb 22 2015

( {A038611_upto(N,M=1)=select( is_A052405, primes([M,N]))} )(350)

next_A038611(n)={until((n=nextprime(n+1))==n=next_A052405(n-1),);n}
( {A038611_vec(n,M=2)=M--;vector(n,i,M=next_A038611(M))} )(20, 1000)
\\ Get 20 terms >= 1000. See also OEIS wiki page. - M. F. Hasler, Jan 14 2020

from sympy import isprime
i=j=1
while j<=5000:
    if isprime(i) and "3" not in str(i):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 07 2017

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

Offset corrected by Arkadiusz Wesolowski, Aug 07 2011

A038617 Primes not containing the digit '9'.

Original entry on oeis.org

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

cp's OEIS Frontend

A375533 a(n) = numerator of Sum_{i=1..n} 1/A038603(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A375534 a(n) = denominator of Sum_{i=1..n} 1/A038603(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A038618 Primes not containing the digit '0'.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

PARI

Python

Formula

A038615 Primes not containing the digit '7'.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A052383 Numbers without 1 as a digit.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

Python

Scala

sh

Formula

Extensions

A038611 Primes not containing the digit '3'.

Views

Author

Keywords

Comments

Links

Crossrefs