A038616 -id:A038616 - 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

A386326 Primes without {0, 8} as digits.

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, 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, 311, 313, 317, 331, 337

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038616 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 3, 4, 5, 6, 7, 9]];

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

primes_with(, 1, [1, 2, 3, 4, 5, 6, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("12345679"), 41))) # uses function/imports in A385776

A386334 Primes without {1, 8} as digits.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 97, 223, 227, 229, 233, 239, 257, 263, 269, 277, 293, 307, 337, 347, 349, 353, 359, 367, 373, 379, 397, 409, 433, 439, 443, 449, 457, 463, 467, 479, 499, 503, 509, 523, 547, 557, 563, 569, 577, 593, 599, 607

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038603 and A038616.

Cf. A000040, A020456, A154761, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 2, 3, 4, 5, 6, 7, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 1] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 2, 3, 4, 5, 6, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("02345679"), 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

A386343 Primes without {3, 8} as digits.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 29, 41, 47, 59, 61, 67, 71, 79, 97, 101, 107, 109, 127, 149, 151, 157, 167, 179, 191, 197, 199, 211, 227, 229, 241, 251, 257, 269, 271, 277, 401, 409, 419, 421, 449, 457, 461, 467, 479, 491, 499, 509, 521, 541, 547, 557, 569, 571, 577, 599

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038611 and A038616.

Cf. A000040, A020464, A385776.

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

Select[Prime[Range[120]], DigitCount[#, 10, 3] == 0 && DigitCount[#, 10, 8] == 0 &]

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

print(list(islice(primes_with("01245679"), 41))) # uses function/imports in A385776

A386348 Primes without {4, 8} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 71, 73, 79, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 293, 307, 311, 313, 317, 331

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038612 and A038616.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 5, 6, 7, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 4] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 5, 6, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("01235679"), 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

A386355 Primes without {6, 8} as digits.

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, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 173, 179, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 277, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038614 and A038616.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 5, 7, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 6] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 5, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("01234579"), 41))) # uses function/imports in A385776

A386357 Primes without {7, 8} as digits.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 23, 29, 31, 41, 43, 53, 59, 61, 101, 103, 109, 113, 131, 139, 149, 151, 163, 191, 193, 199, 211, 223, 229, 233, 239, 241, 251, 263, 269, 293, 311, 313, 331, 349, 353, 359, 401, 409, 419, 421, 431, 433, 439, 443, 449, 461, 463, 491, 499, 503

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038615 and A038616.

Cf. A000040, A020470, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 5, 6, 9]];

Select[Prime[Range[120]], DigitCount[#, 10, 7] == 0 && DigitCount[#, 10, 8] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 5, 6, 9]) \\ uses function in A385776

print(list(islice(primes_with("01234569"), 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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A386326 Primes without {0, 8} as digits.

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Magma

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A386334 Primes without {1, 8} as digits.

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Magma

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A386338 Primes without {2, 8} as digits.

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Magma

Maple

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A386343 Primes without {3, 8} as digits.

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Magma

Mathematica

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A386348 Primes without {4, 8} as digits.

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Magma

Mathematica

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A386352 Primes without {5, 8} as digits.

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Magma

Mathematica

PARI