A034845 -id:A034845 - cp's OEIS frontend

A235154 Primes which have one or more occurrences of exactly two different digits.

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811

Offset: 1

Colin Barker, Jan 04 2014

The first term having a repeated digit is 101.

a(3402) > 10^10.

Michael S. Branicky, Table of n, a(n) for n = 1..12000 (terms 651..3401 from Christopher M. Conrey, terms 1..650 from Colin Barker)
David A. Corneth, PARI program

Cf. A034845, A235155-A235161, A030291.

s=[]; forprime(n=10, 1000, if(#vecsort(eval(Vec(Str(n))),,8)==2, s=concat(s, n))); s

is(n)=isprime(n) && #Set(digits(n))==2 \\ Charles R Greathouse IV, Feb 23 2017

PARI
```
\\ See Corneth link
```

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
from itertools import count, islice, combinations_with_replacement, product
def agen():
    for digits in count(2):
        s = set()
        for pair in product("0123456789", "1379"):
            if pair[0] == pair[1]: continue
            for c in combinations_with_replacement(pair, digits):
                if len(set(c)) < 2 or sum(int(ci) for ci in c)%3 == 0:
                    continue
                for p in multiset_permutations(c):
                    if p[0] == "0": continue
                    t = int("".join(p))
                    if isprime(t):
                        s.add(t)
        yield from sorted(s)
print(list(islice(agen(), 100))) # Michael S. Branicky, Jan 23 2022

A062353 Primes of the form bbbbba... where a and b are 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, 83, 89, 97, 113, 223, 227, 229, 331, 337, 443, 449, 557, 661, 773, 881, 883, 887, 991, 997, 1117, 2221, 3331, 4441, 4447, 5557, 6661, 8887, 11113, 11117, 11119, 22229

Offset: 1

Amarnath Murthy, Jun 23 2001

Number of terms of n digits: 4, 21, 16, 8, 9, 8, 2, 8, 7, 3, 4, 5, 2, 2, 4, 0, 3, 4, 2, 3, 2, 2, 4, 1, 0, ..., . - Robert G. Wilson v, May 29 2011

			4441 is a member where a=1 and b = 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..260

Cf. A034845, A061022.

f[n_] := Select[ Union@ Flatten@ Table[ FromDigits@ Join[ Table[b, {n - 1}], {a}], {b, 9}, {a, {1, 3, 7, 9}}], PrimeQ]; f[1] = {2, 3, 5, 7}; Array[f, 5] // Flatten (* Robert G. Wilson v, May 29 2011 *)

print1("2, 3, 5, 7");for(n=2,20,forstep(k=10^n\9-1,10^n-9,10^n\9-1,for(m=k+1,k+9,if(isprime(m),print1(", "m))))) \\ Charles R Greathouse IV, May 29 2011

More terms from Jason Earls, Jun 26 2001

Corrected and extended by Dean Hickerson, Jul 10 2001

A061022 Primes of the form abbbbb... where a and b are 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, 83, 89, 97, 199, 211, 233, 277, 311, 433, 499, 577, 599, 677, 733, 811, 877, 911, 977, 1777, 1999, 2111, 2333, 2777, 2999, 4111, 4999, 5333, 7333, 8111, 8999, 23333, 47777

Offset: 1

Amarnath Murthy, Jun 23 2001

Number of terms of n digits: 4, 21, 15, 12, 7, 8, 2, 7, 2, 3, 5, 2, 2, 7, 2, 4, 2, 2, 4, 3, 1, 0, 3, 3, 0, ..., . - Robert G. Wilson v, May 29 2011

			4111 is a member where a=4 and b = 1.

Robert G. Wilson v, Table of n, a(n) for n = 1..262 (first 79 terms from Harry J. Smith) (shortened by _N. J. A. Sloane_, Jan 13 2019)

Cf. A034845, A062353.

N:= 20:
A:= 2,3,5,7:
for n from 2 to N do
  for a from 1 to 9 do
    for b in [1,3,7,9] do
       p:= a*10^(n-1) + b*(10^(n-1)-1)/9;
       if isprime(p) then A:= A,p fi
    od
  od
od:
A;  # Robert Israel, Oct 13 2014

f[n_] := Select[ Union@ Flatten@ Table[ FromDigits@ Join[{a}, Table[b, {n - 1}]], {a, 9}, {b, {1, 3, 7, 9}}], PrimeQ]; Array[f, 5] // Flatten (* Robert G. Wilson v, May 29 2011 *)

{ n=r=0; default(primelimit, 1777777777); forprime (p=2, 1777777777, if (p>100, r=p\10; d=p-r*10; while (r>9 && r-r\10*10 == d, r\=10)); if (r<=9, write("b061022.txt", n++, " ", p)) ) } \\ Harry J. Smith, Jul 16 2009

Corrected and extended by Dean Hickerson, Jul 10 2001

A385481 Primes whose decimal expansion consists of the concatenation of ij, iijj, iiijjj,..., and m i’s followed by m j’s, i != j, where 1<= i, j <= 9 and m > 0.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 31331133311133331111, 37337733377733337777, 43443344433344443333, 49449944499944449999

Offset: 1

Gonzalo Martínez, Jun 30 2025

a(25) has 812 digits and starts with 292299222999...999, where the last concatenated strings have 28 2's followed by 28 9's.
Since each term is prime, then j = 1, 3, 7, and 9 and i + j == 1, 2 (mod 3).
m == 1 (mod 3). Proof. If k is a term whose last concatenated string has m i's followed by m j's, then it has 2(1 + 2 + ... + m) = m(m + 1) digits, whose sum is (i + j)*m(m + 1)/2, so that if m == 0, 2 (mod 3), then the sum of digits of k is a multiple of 3 and so k is not prime.
Are there infinite primes of this form?
From Michael S. Branicky, Jul 01 2025: (Start)
A probabilistic argument suggests the sequence is finite.
a(26), if it exists, has m > 321 and > 103362 digits. (End)

			for i = 3, j = 1 and m = 4, by concatenating 31, 3311, 333111, 33331111 the prime 31331133311133331111 is obtained.

Gonzalo Martínez, Table of n, a(n) for n = 1..25

Cf. A059170, A034845.

from gmpy2 import is_prime, mpz
from itertools import count, islice, product
def agen(): yield from (p for m in count(1) for i in "123456789" for j in "1379" if i != j and is_prime(p:=int(mpz("".join(i*k+j*k for k in range(1, m+1))))))
print(list(islice(agen(), 24))) # Michael S. Branicky, Jun 30 2025

A385637 Primes whose decimal expansion consists of the concatenation of m i’s followed by m j’s, ..., iiijjj, iijj and ij, i != j, where 1 <= i, j <= 9 and m > 0.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 44443333444333443343, 55555553333333555555333333555553333355553333555333553353

Offset: 1

Gonzalo Martínez, Jul 05 2025

Similar to A385481, but the blocks of i's and j's are concatenated from longest length to shortest, where a(n) contains terms of a length not recorded in A385481, such as length 56, 182 and 272.
a(23) has 182 digits and starts with 13 2's followed by 13 9's.
a(24) has 272 digits and starts with 16 9's followed by 16 7's.
a(25), if it exists, has m > 200 and > 40200 digits.

			For i = 4, j = 3 and m = 4, by concatenating 44443333, 444333, 4433 and 43 the prime 44443333444333443343 is obtained.

Gonzalo Martínez, Table of n, a(n) for n = 1..24

Cf. A385481, A059170, A034845.

cp's OEIS Frontend

A235154 Primes which have one or more occurrences of exactly two different digits.

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PARI

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A062353 Primes of the form bbbbba... where a and b are digits.

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Mathematica

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A061022 Primes of the form abbbbb... where a and b are digits.

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Maple

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A385481 Primes whose decimal expansion consists of the concatenation of ij, iijj, iiijjj,..., and m i’s followed by m j’s, i != j, where 1<= i, j <= 9 and m > 0.

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Programs

Python

A385637 Primes whose decimal expansion consists of the concatenation of m i’s followed by m j’s, ..., iiijjj, iijj and ij, i != j, where 1 <= i, j <= 9 and m > 0.

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Author

Keywords

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Examples

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