A038611 -id:A038611 - cp's OEIS frontend

A212525 Primes containing a digit 3.

3, 13, 23, 31, 37, 43, 53, 73, 83, 103, 113, 131, 137, 139, 163, 173, 193, 223, 233, 239, 263, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 431, 433, 439, 443, 463, 503, 523, 563, 593, 613, 631, 643, 653, 673, 683

Offset: 1

Jaroslav Krizek, Jun 12 2012

Supersequence of A045709, A106103 and A106099. Subsequence of A011533 and A062667.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A045709 (primes with first digit 3), A011533 (numbers containing a digit 3), A062667 (numbers n such that every divisor of n (except 1) contains the digit 3), A106099 (primes with maximal digit = 3), A106103 (primes with minimal digit = 3), A038611 (primes not containing digit 3).

Select[Prime[Range[200]], MemberQ[IntegerDigits[#], 3] &] (* T. D. Noe, Jun 12 2012 *)

a(n) ~ n log n. - Charles R Greathouse IV, Nov 01 2022

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

A380906 Primes without {3, 5} as digits.

Original entry on oeis.org

2, 7, 11, 17, 19, 29, 41, 47, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 149, 167, 179, 181, 191, 197, 199, 211, 227, 229, 241, 269, 271, 277, 281, 401, 409, 419, 421, 449, 461, 467, 479, 487, 491, 499, 601, 607, 617, 619, 641, 647, 661, 677, 691, 701, 709, 719, 727, 761, 769, 787, 797

Offset: 1

Vincenzo Librandi, Feb 09 2025

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 3275 terms from Vincenzo Librandi)
Index to entries for primes with digits in a given set

Intersection of A038611 and A038613.

Cf. A000040, A020462, A329760.

[p: p in PrimesUpTo(700) | not 3 in Intseq(p) and not 5 in Intseq(p) ];

Select[Prime[Range[120]],DigitCount[#,10,3]==0&&DigitCount[#,10,5]==0&]

isok(p) = if (isprime(p), my(d=digits(p)); (#select(x->(x==3), d)==0) && (#select(x->(x==5), d)==0)); \\ Michel Marcus, Feb 10 2025

from itertools import count, islice
from sympy import isprime
def A380906_gen(): # generator of terms
    return filter(isprime,(int(oct(n)[2:].translate({51:52,52:54,53:55,54:56,55:57})) for n in count(1)))
A380906_list = list(islice(A380906_gen(),20)) # Chai Wah Wu, Feb 12 2025

A224320 Primes without "3" as a digit that remain prime when any single digit is replaced with "3".

Original entry on oeis.org

2, 5, 7, 11, 17, 41, 47, 71, 107, 167, 179, 197, 449, 859, 1019, 1061, 1499, 2089, 16901, 47717, 56269, 86269, 11917049

Offset: 1

Arkadiusz Wesolowski, Apr 03 2013

No more terms < 10^13.

Carlos Rivera, Puzzle 591

Cf. A224319, A224321-A224322. Subsequence of A038611.

lst = {}; n = 3; Do[If[PrimeQ[p], i = IntegerDigits[p]; If[FreeQ[i, n], t = 0; s = IntegerLength[p]; Do[If[PrimeQ@FromDigits@Insert[Drop[i, {d}], n, d], t++, Break[]], {d, s}]; If[t == s, AppendTo[lst, p]]]], {p, 86269}]; lst
p3Q[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,3] && AllTrue[ FromDigits/@ Table[ReplacePart[idn,i->3],{i,IntegerLength[n]}],PrimeQ]]; Select[Prime[Range[10^6]],p3Q] (* The program uses the function AllTrue from Mathematica version 10 *) (* Harvey P. Dale, Aug 20 2014 *)

A386321 Primes without {0, 3} as digits.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 41, 47, 59, 61, 67, 71, 79, 89, 97, 127, 149, 151, 157, 167, 179, 181, 191, 197, 199, 211, 227, 229, 241, 251, 257, 269, 271, 277, 281, 419, 421, 449, 457, 461, 467, 479, 487, 491, 499, 521, 541, 547, 557, 569, 571, 577, 587, 599, 617

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038611 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 4, 5, 6, 7, 8, 9]];

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

primes_with(, 1, [1, 2, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("12456789"), 41))) # uses function/imports in A385776

A386329 Primes without {1, 3} as digits.

Original entry on oeis.org

2, 5, 7, 29, 47, 59, 67, 79, 89, 97, 227, 229, 257, 269, 277, 409, 449, 457, 467, 479, 487, 499, 509, 547, 557, 569, 577, 587, 599, 607, 647, 659, 677, 709, 727, 757, 769, 787, 797, 809, 827, 829, 857, 859, 877, 887, 907, 929, 947, 967, 977, 997, 2027, 2029, 2069

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038611.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 4, 5, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 3] == 0 &]

primes_with(, 1, [0, 2, 4, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02456789"), 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

A386340 Primes without {3, 4} as digits.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 59, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 151, 157, 167, 179, 181, 191, 197, 199, 211, 227, 229, 251, 257, 269, 271, 277, 281, 509, 521, 557, 569, 571, 577, 587, 599, 601, 607, 617, 619, 659, 661, 677, 691, 701, 709, 719, 727, 751

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038611 and A038612.

Cf. A000040, A020461, A380906, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 5, 6, 7, 8, 9]];

f:= n-> (l-> add([$0..2, $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, 3] == 0 && DigitCount[#, 10, 4] == 0 &]

primes_with(, 1, [0, 1, 2, 5, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01256789"), 41))) # uses function/imports in A385776

A386341 Primes without {3, 6} as digits.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 41, 47, 59, 71, 79, 89, 97, 101, 107, 109, 127, 149, 151, 157, 179, 181, 191, 197, 199, 211, 227, 229, 241, 251, 257, 271, 277, 281, 401, 409, 419, 421, 449, 457, 479, 487, 491, 499, 509, 521, 541, 547, 557, 571, 577, 587, 599, 701, 709

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038611 and A038614.

Cf. A000040, A380906,A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 4, 5, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 3] == 0 && DigitCount[#, 10, 6] == 0 &]

primes_with(, 1, [0, 1, 2, 4, 5, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01245789"), 41))) # uses function/imports in A385776

A386342 Primes without {3, 7} as digits.

Original entry on oeis.org

2, 5, 11, 19, 29, 41, 59, 61, 89, 101, 109, 149, 151, 181, 191, 199, 211, 229, 241, 251, 269, 281, 401, 409, 419, 421, 449, 461, 491, 499, 509, 521, 541, 569, 599, 601, 619, 641, 659, 661, 691, 809, 811, 821, 829, 859, 881, 911, 919, 929, 941, 991, 1009, 1019

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038611 and A038615.

Cf. A000040, A020463, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 4, 5, 6, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 3] == 0 && DigitCount[#, 10, 7] == 0 &]

primes_with(, 1, [0, 1, 2, 4, 5, 6, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01245689"), 41))) # uses function/imports in A385776

cp's OEIS Frontend

A212525 Primes containing a digit 3.

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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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Maple

Mathematica

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A380906 Primes without {3, 5} as digits.

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Magma

Mathematica

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A224320 Primes without "3" as a digit that remain prime when any single digit is replaced with "3".

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Mathematica

A386321 Primes without {0, 3} as digits.

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Magma

Mathematica

PARI

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A386329 Primes without {1, 3} as digits.

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Magma

Mathematica

PARI

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A386335 Primes without {2, 3} as digits.

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Magma

Mathematica

PARI

Python

A386340 Primes without {3, 4} as digits.

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