A038617 -id:A038617 - cp's OEIS frontend

A224322 Primes without "9" as a digit that remain prime when any single digit is replaced with "9".

17, 107, 137, 347, 1013, 1433, 1613, 3767, 4337, 8623, 25633, 31114073

Offset: 1

Arkadiusz Wesolowski, Apr 03 2013

No more terms < 10^13.

Carlos Rivera, Puzzle 591

Cf. A224319-A224321. Subsequence of A038617.

lst = {}; n = 9; 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, 25633}]; lst
pr9Q[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,9]&&AllTrue[Table[ FromDigits[ ReplacePart[ idn,i->9]],{i,Length[idn]}],PrimeQ]]; Select[ Prime[Range[2*10^6]],pr9Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 30 2015 *)

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

A386358 Primes without {7, 9} as digits.

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 61, 83, 101, 103, 113, 131, 151, 163, 181, 211, 223, 233, 241, 251, 263, 281, 283, 311, 313, 331, 353, 383, 401, 421, 431, 433, 443, 461, 463, 503, 521, 523, 541, 563, 601, 613, 631, 641, 643, 653, 661, 683, 811, 821, 823

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038615 and A038617.

Cf. A000040, A020471, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 4, 5, 6, 8]];

Select[Prime[Range[120]], DigitCount[#, 10, 7] == 0 && DigitCount[#, 10, 9] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 4, 5, 6, 8]) \\ uses function in A385776

print(list(islice(primes_with("01234568"), 41))) # uses function/imports in A385776

A036962 Primes without {8, 9} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 101, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 211, 223, 227, 233, 241, 251, 257, 263, 271, 277, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 401, 421, 431, 433, 443, 457, 461

Offset: 1

Patrick De Geest, Jan 04 1999

Harvey P. Dale, Table of n, a(n) for n = 1..10000
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

Subsequence of A038617.

Cf. A036952-A036964.

Select[FromDigits/@Tuples[Range[0,7],3],PrimeQ] (* Harvey P. Dale, Apr 13 2017 *)

A360147 Primes in base 10 that are also prime when read in a smaller base that is one plus the largest digit in the prime in base 10.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 31, 37, 43, 61, 73, 101, 103, 107, 113, 131, 151, 223, 227, 233, 241, 251, 277, 307, 311, 331, 337, 373, 401, 461, 463, 467, 521, 547, 557, 577, 661, 673, 701, 827, 887, 1013, 1033, 1103, 1151, 1181, 1213, 1223, 1231, 1301, 1327, 1567

Offset: 1

Alfred Jacob Mohan, Jan 27 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequence of A038617.

q:= n-> isprime(n) and (l-> (d-> d<9 and isprime(add(l[i]*
   (d+1)^(i-1), i=1..nops(l))))(max(l)))(convert(n, base, 10)):
select(q, [$1..2000])[];  # Alois P. Heinz, Jan 27 2023

q[p_] := Module[{d = IntegerDigits[p], b}, b = Max[d] + 1; b <= 9 && PrimeQ[FromDigits[d, b]]]; Select[Prime[Range[250]], q] (* Amiram Eldar, Jan 27 2023 *)

isok(p)=if(isprime(p), my(v=digits(p), b=vecmax(v)+1); b<10 && isprime(fromdigits(v,b)), 0) \\ Andrew Howroyd, Jan 27 2023

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    b = int(max(s)) + 1
    return b != 10 and isprime(int(s, b))
print([k for k in range(1600) if ok(k)]) # Michael S. Branicky, Jan 27 2023

More terms from Michael S. Branicky, Jan 27 2023

A386327 Primes without {0, 9} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 83, 113, 127, 131, 137, 151, 157, 163, 167, 173, 181, 211, 223, 227, 233, 241, 251, 257, 263, 271, 277, 281, 283, 311, 313, 317, 331, 337, 347, 353, 367, 373, 383, 421, 431, 433, 443, 457, 461

Offset: 1

Jason Bard, Jul 19 2025

Intersection of A038617 and A038618.

Cf. A000040, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [1, 2, 3, 4, 5, 6, 7, 8]];

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

primes_with(, 1, [1, 2, 3, 4, 5, 6, 7, 8]) \\ uses function in A385776

print(list(islice(primes_with("12345678"), 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

A386344 Primes without {3, 9} as digits.

Original entry on oeis.org

2, 5, 7, 11, 17, 41, 47, 61, 67, 71, 101, 107, 127, 151, 157, 167, 181, 211, 227, 241, 251, 257, 271, 277, 281, 401, 421, 457, 461, 467, 487, 521, 541, 547, 557, 571, 577, 587, 601, 607, 617, 641, 647, 661, 677, 701, 727, 751, 757, 761, 787, 811, 821, 827, 857

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038611 and A038617.

Cf. A000040, A385776.

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

Select[Prime[Range[120]], DigitCount[#, 10, 3] == 0 && DigitCount[#, 10, 9] == 0 &]

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

print(list(islice(primes_with("01245678"), 41))) # uses function/imports in A385776

A386349 Primes without {4, 9} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 61, 67, 71, 73, 83, 101, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 181, 211, 223, 227, 233, 251, 257, 263, 271, 277, 281, 283, 307, 311, 313, 317, 331, 337, 353, 367, 373, 383, 503, 521, 523, 557, 563, 571

Offset: 1

Jason Bard, Jul 20 2025

Intersection of A038612 and A038617.

Cf. A000040, A020466, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 1, 2, 3, 5, 6, 7, 8]];

Select[Prime[Range[120]], DigitCount[#, 10, 4] == 0 && DigitCount[#, 10, 9] == 0 &]

primes_with(, 1, [0, 1, 2, 3, 5, 6, 7, 8]) \\ uses function in A385776

print(list(islice(primes_with("01235678"), 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

A224322 Primes without "9" as a digit that remain prime when any single digit is replaced with "9".

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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

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A386358 Primes without {7, 9} as digits.

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Magma

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A036962 Primes without {8, 9} as digits.

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Mathematica

A360147 Primes in base 10 that are also prime when read in a smaller base that is one plus the largest digit in the prime in base 10.

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Maple

Mathematica

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A386327 Primes without {0, 9} as digits.

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Magma

Mathematica

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A386339 Primes without {2, 9} as digits.

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Magma

Mathematica

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A386344 Primes without {3, 9} as digits.

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