A038604 -id:A038604 - cp's OEIS frontend

A052404 Numbers without 2 as a digit.

0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Henry Bottomley, Mar 13 2000

M. J. Halm, Word Weirdness, Mpossibilities 66 (Feb. 1998), p. 5.

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

Cf. A004177, A004721, A072809, A082831 (Kempner series).
Cf. A052382 (without 0), A052383 (without 1), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).
See A038604 for the subset of primes. - M. F. Hasler, Jan 11 2020

a052404 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + if r > 1 then r + 1 else r  where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014

[ n: n in [0..89] | not 2 in Intseq(n) ];  // Bruno Berselli, May 28 2011

a:= proc(n) local l, m; l, m:= 0, n-1;
      while m>0 do l:= (d->
        `if`(d<2, d, 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

ban2Q[n_]:=FreeQ[IntegerDigits[n],2]==True; Select[Range[0,89],ban2Q[#] &] (* Jayanta Basu, May 17 2013 *)
Select[Range[0,100],DigitCount[#,10,2]==0&] (* Harvey P. Dale, Apr 13 2015 *)

lista(nn, d=2) = {for (n=0, nn, if (!vecsearch(vecsort(digits(n),,8),d), print1(n, ", ")););} \\ Michel Marcus, Feb 21 2015

apply( {A052404(n)=fromdigits(apply(d->d+(d>1),digits(n-1,9)))}, [1..99])
next_A052404(n, d=digits(n+=1))={for(i=1, #d, d[i]==2&&return((1+n\d=10^(#d-i))*d)); n} \\ least a(k) > n: if there's a digit 2 in n+1, replace the first occurrence by 3 and all following digits by 0.
(A052404_vec(N)=vector(N, i, N=if(i>1, next_A052404(N))))(99) \\ first N terms
select( {is_A052404(n)=!setsearch(Set(digits(n)),2)}, [0..99])
(A052404_upto(N)=select( is_A052404, [0..N]))(99) \\ M. F. Hasler, Jan 11 2020

from gmpy2 import digits
def A052404(n): return int(''.join(str(int(d)+1) if d>'1' else d for d in digits(n-1,9))) # Chai Wah Wu, Aug 30 2024

seq 0 1000 | grep -v 2; # Joerg Arndt, May 29 2011

If the offset were changed to 0: a(0) = 0, a(n+1) = f(a(n)+1,a(n)+1) where f(x,y) = if x<10 and x<>2 then y else if x mod 10 = 2 then f(y+1,y+1) else f(floor(x/10),y). - Reinhard Zumkeller, Mar 02 2008
a(n) = replace digits d > 1 by d + 1 in base-9 representation of n - 1. - Reinhard Zumkeller, Oct 07 2014
Sum_{k>1} 1/a(k) = A082831 = 19.257356... (Kempner series). - Bernard Schott, Jan 12 2020, edited by M. F. Hasler, Jan 14 2020

Offset changed by Reinhard Zumkeller, Oct 07 2014

A329760 Primes without {2, 7} as digits.

Original entry on oeis.org

3, 5, 11, 13, 19, 31, 41, 43, 53, 59, 61, 83, 89, 101, 103, 109, 113, 131, 139, 149, 151, 163, 181, 191, 193, 199, 311, 313, 331, 349, 353, 359, 383, 389, 401, 409, 419, 431, 433, 439, 443, 449, 461, 463, 491, 499, 503, 509, 541, 563, 569, 593, 599, 601, 613

Offset: 1

Alois P. Heinz, Nov 20 2019

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Marianne Freiberger, Primes without 7s.
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, A020459, A038604, A038615.

[p: p in PrimesUpTo(700) | not 2 in Intseq(p) and not 7 in Intseq(p) ]; // Vincenzo Librandi, Jan 02 2020

Select[Prime[Range[120]], DigitCount[#, 10, 2]==0 && DigitCount[#, 10, 7]==0 &] (* Vincenzo Librandi, Jan 02 2020 *)

{ A038604 } intersect { A038615 }.

A254315 Number of distinct digits in the prime factorization of n (counting terms of the form p^1 as p).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3

Offset: 2

Michel Lagneau, Jan 28 2015

Write n as product of primes raised to powers; then a(n) is the total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1).
a(n)<=10. The least n such that a(n)=10 is n = 41701690 = 2*5*47*83*1069.
Property: a(p) = A043537(p), for p prime.
From Michel Marcus, Feb 21 2015: (Start)
For p in A038604, a(p^2) = A043537(p) + 1.
For p in A038611, a(p^3) = A043537(p) + 1.
For p in A038612, a(p^4) = A043537(p) + 1.
For p in A038613, a(p^5) = A043537(p) + 1.
For p in A038614, a(p^6) = A043537(p) + 1.
For p in A038615, a(p^7) = A043537(p) + 1.
For p in A038616, a(p^8) = A043537(p) + 1.
For p in A038617, a(p^9) = A043537(p) + 1.
(End)

			a(36)=2 because 36 = 2^2 * 3^2 => 2 distinct digits.
a(414)=2 because 414 = 2 * 3^2 * 23 => 2 distinct digits.

Michel Lagneau, Table of n, a(n) for n = 2..10000

Cf. A027748, A043537, A050252, A254317.

with(ListTools):
nn:=100:
  for n from 2 to nn do:
    n0:=length(n):lst:={}:x0:=ifactors(n):
    y:=Flatten(x0[2]):z:=convert(y,set):
    z1:=z minus {1}:nn0:=nops(z1):
     for k from 1 to nn0 do :
      t1:=convert(z1[k],base,10):z2:=convert(t1,set):
      lst:=lst union z2:
     od:
     nn1:=nops(lst):printf(`%d, `,nn1):
     od :

f[n_] := Block[{pf = FactorInteger@ n, i}, Length@ DeleteDuplicates@ Flatten@ IntegerDigits@ Rest@ Flatten@ Reap@ Do[If[Last[pf[[i]]] == 1, Sow@ First@ pf[[i]], Sow@ FromDigits@ Flatten[IntegerDigits /@ pf[[i]]]], {i, Length@ pf}]]; Array[f,100] (* Michael De Vlieger, Jan 29 2015 *)

print1(1,", ");for(k=2,100,s=[];F=factor(k);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));print1(#vecsort(s,,8),", ")) \\ Derek Orr, Jan 30 2015

from sympy import factorint
def A254315(n):
    return len(set([x for l in [[d for d in str(p)]+[d for d in str(e) if d != '1'] for p,e in factorint(n).items()] for x in l]))
# Chai Wah Wu, Feb 24 2015

A386320 Primes without {0, 2} as digits.

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, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 419, 431, 433, 439

Offset: 1

Jason Bard, Jul 18 2025

Intersection of A038604 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 3, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 0] == 0 && DigitCount[#, 10, 2] == 0 &]

primes_with(, 1, [1, 3, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("13456789"), 41))) # uses function/imports in A385776

A386328 Primes without {1, 2} as digits.

Original entry on oeis.org

3, 5, 7, 37, 43, 47, 53, 59, 67, 73, 79, 83, 89, 97, 307, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 487, 499, 503, 509, 547, 557, 563, 569, 577, 587, 593, 599, 607, 643, 647, 653, 659, 673, 677, 683, 709

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038604.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 3, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 2] == 0 &]

primes_with(, 1, [0, 3, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("03456789"), 41))) # uses function/imports in A385776

A386335 Primes without {2, 3} as digits.

Original entry on oeis.org

5, 7, 11, 17, 19, 41, 47, 59, 61, 67, 71, 79, 89, 97, 101, 107, 109, 149, 151, 157, 167, 179, 181, 191, 197, 199, 401, 409, 419, 449, 457, 461, 467, 479, 487, 491, 499, 509, 541, 547, 557, 569, 571, 577, 587, 599, 601, 607, 617, 619, 641, 647, 659, 661, 677, 691

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038604 and A038611.

Cf. A000040, A020458, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 2] == 0 && DigitCount[#, 10, 3] == 0 &]

primes_with(, 1, [0, 1, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01456789"), 41))) # uses function/imports in A385776

A386336 Primes without {2, 4} as digits.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 307, 311, 313, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 503, 509, 557, 563, 569

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038604 and A038612.

Cf. A000040, A361822, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 3, 5, 6, 7, 8, 9]];

f:= n-> (l-> add([0, 1, 3, $5..9][l[j]+1]*10^(j-1), j=1..nops(l)))(convert(n, base, 8)):
select(isprime, [seq(f(i), i=0..600)])[];  # Alois P. Heinz, Jul 19 2025

Select[Prime[Range[120]], DigitCount[#, 10, 2] == 0 && DigitCount[#, 10, 4] == 0 &]

primes_with(, 1, [0, 1, 3, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01356789"), 41))) # uses function/imports in A385776

A386337 Primes without {2, 6} as digits.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 373, 379, 383, 389, 397, 401, 409, 419, 431

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038604 and A038614.

Cf. A000040, A329760, A361822, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 3, 4, 5, 7, 8, 9]];

f:= n-> (l-> add([0, 1, $3..5, $7..9][l[j]+1]*10^(j-1), j=1..nops(l)))(convert(n, base, 8)):
select(isprime, [seq(f(i), i=0..600)])[];  # Alois P. Heinz, Jul 19 2025

Select[Prime[Range[120]], DigitCount[#, 10, 2] == 0 && DigitCount[#, 10, 6] == 0 &]

primes_with(, 1, [0, 1, 3, 4, 5, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01345789"), 41))) # uses function/imports in A385776

A386338 Primes without {2, 8} as digits.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 397, 401, 409, 419, 431

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038604 and A038616.

Cf. A000040, A329760, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 3, 4, 5, 6, 7, 9]];

f:= n-> (l-> add([0, 1, $3..7, 9][l[j]+1]*10^(j-1), j=1..nops(l)))(convert(n, base, 8)):
select(isprime, [seq(f(i), i=0..600)])[];  # Alois P. Heinz, Jul 19 2025

Select[Prime[Range[120]], DigitCount[#, 10, 2] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 1, 3, 4, 5, 6, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("01345679"), 41))) # uses function/imports in A385776

A386339 Primes without {2, 9} as digits.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 83, 101, 103, 107, 113, 131, 137, 151, 157, 163, 167, 173, 181, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 383, 401, 431, 433, 443, 457, 461, 463, 467, 487, 503, 541, 547, 557, 563, 571, 577

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038604 and A038617.

Cf. A000040, A020460, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 3, 4, 5, 6, 7, 8]];

Select[Prime[Range[120]], DigitCount[#, 10, 2] == 0 && DigitCount[#, 10, 9] == 0 &]

primes_with(, 1, [0, 1, 3, 4, 5, 6, 7, 8]) \\ uses function in A385776

print(list(islice(primes_with("01345678"), 41))) # uses function/imports in A385776

cp's OEIS Frontend

A052404 Numbers without 2 as a digit.

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A329760 Primes without {2, 7} as digits.

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A254315 Number of distinct digits in the prime factorization of n (counting terms of the form p^1 as p).

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A386320 Primes without {0, 2} as digits.

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A386328 Primes without {1, 2} as digits.

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A386335 Primes without {2, 3} as digits.

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A386336 Primes without {2, 4} as digits.

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