A038613 -id:A038613 - cp's OEIS frontend

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

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

A386323 Primes without {0, 5} as digits.

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, 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, 311, 313, 317, 331, 337, 347

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038613 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 3, 4, 6, 7, 8, 9]];

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

primes_with(, 1, [1, 2, 3, 4, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("12346789"), 41))) # uses function/imports in A385776

A386331 Primes without {1, 5} as digits.

Original entry on oeis.org

2, 3, 7, 23, 29, 37, 43, 47, 67, 73, 79, 83, 89, 97, 223, 227, 229, 233, 239, 263, 269, 277, 283, 293, 307, 337, 347, 349, 367, 373, 379, 383, 389, 397, 409, 433, 439, 443, 449, 463, 467, 479, 487, 499, 607, 643, 647, 673, 677, 683, 709, 727, 733, 739, 743, 769

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038613.

Cf. A000040, A020453, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 3, 4, 6, 7, 8, 9]];

f:= n-> (l-> add([0, $2..4, $6..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, 1] == 0 && DigitCount[#, 10, 5] == 0 &]

primes_with(, 1, [0, 2, 3, 4, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("02346789"), 41))) # uses function/imports in A385776

A386345 Primes without {4, 5} as digits.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 367

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038612 and A038613.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 6, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 4] == 0 && DigitCount[#, 10, 5] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 6, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01236789"), 41))) # uses function/imports in A385776

A386350 Primes without {5, 6} as digits.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038613 and A038614.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 7, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 5] == 0 && DigitCount[#, 10, 6] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01234789"), 41))) # uses function/imports in A385776

A386351 Primes without {5, 7} as digits.

Original entry on oeis.org

2, 3, 11, 13, 19, 23, 29, 31, 41, 43, 61, 83, 89, 101, 103, 109, 113, 131, 139, 149, 163, 181, 191, 193, 199, 211, 223, 229, 233, 239, 241, 263, 269, 281, 283, 293, 311, 313, 331, 349, 383, 389, 401, 409, 419, 421, 431, 433, 439, 443, 449, 461, 463, 491, 499

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038613 and A038615.

Cf. A000040, A020467, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 6, 8, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 5] == 0 && DigitCount[#, 10, 7] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 6, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("01234689"), 41))) # uses function/imports in A385776

A386352 Primes without {5, 8} as digits.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 163, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 293, 307, 311, 313, 317, 331, 337, 347

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038613 and A038616.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 6, 7, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 5] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 6, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("01234679"), 41))) # uses function/imports in A385776

A386353 Primes without {5, 9} as digits.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 61, 67, 71, 73, 83, 101, 103, 107, 113, 127, 131, 137, 163, 167, 173, 181, 211, 223, 227, 233, 241, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 337, 347, 367, 373, 383, 401, 421, 431, 433, 443, 461, 463, 467

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038613 and A038617.

Cf. A000040, A020468, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 6, 7, 8]];

Select[Prime[Range[120]], DigitCount[#, 10, 5] == 0 && DigitCount[#, 10, 9] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 6, 7, 8]) \\ uses function in A385776

print(list(islice(primes_with("01234678"), 41))) # uses function/imports in A385776

cp's OEIS Frontend

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

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A380906 Primes without {3, 5} as digits.

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Magma

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A386323 Primes without {0, 5} as digits.

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Magma

Mathematica

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A386331 Primes without {1, 5} as digits.

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Magma

Maple

Mathematica

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A386345 Primes without {4, 5} as digits.

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Magma

Mathematica

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Python

A386350 Primes without {5, 6} as digits.

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Magma

Mathematica

PARI

Python

A386351 Primes without {5, 7} as digits.

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Magma

Mathematica

PARI