A029743 -id:A029743 - cp's OEIS frontend

A132129 Largest prime with distinct digits when written in base n.

2, 19, 19, 577, 7417, 114229, 2053313, 42373937, 987654103, 25678048763, 736867805209, 23136292864193, 789018236128391, 29043982525257901, 1147797409030815779, 48471109094902530293, 2178347851919531491093, 103805969587115219167613, 5228356786703601108008083

Offset: 2

Rick L. Shepherd, Aug 11 2007

a(10) = 987654103 = A007810(9). For n >= 3, a(n) < A062813(n), a multiple of n.
Contribution from R. J. Mathar, May 15 2010: (Start)
Supposed all digits are used and the digits at positions 0 to n-1 are d_0, d_1,... d_{n-1}, the candidates are d_0+d_1*n+d_2*n^2+....+d_{n-1}*n^(n-1).
These values are (n-1)*n/2 (mod n-1), and they cannot be prime if n is even, because this number is = 0 (mod n-1) then, showing that n-1 is a divisor.
In conclusion, if n is even, the entries have at most n-1 digits in base n. (End)
If n is odd then the candidate numbers considered in the previous comment are divisible by (n-1)/2. Hence, we conclude that for n>3, a(n) has at most n-1 digits in base n. Conjecture: for n>3, a(n) has exactly n-1 digits in base n. - Eric M. Schmidt, Oct 26 2014

			a(9) = 42373937 as the prime 42373937 (base 10) = 87654102 (base 9), the largest prime number with distinct digits when represented in base 9.

Eric M. Schmidt, Table of n, a(n) for n = 2..200

Cf. A062813, A007810, A029743.

def a(n) :
    if n==2 : return 2
    if n==3 : return 19
    for P in Permutations(range(n-1,-1,-1), n-1) :
        N = sum(P[-1-i]*n^i for i in range(n-1))
        if is_prime(N) : return N
# Eric M. Schmidt, Oct 26 2014

Removed my claim of finiteness of the sequence. - R. J. Mathar, May 18 2010

a(11)-a(20) from Eric M. Schmidt, Oct 26 2014

A256339 Distinct-digit primes that are concatenation of prime(m) and m for some m.

Original entry on oeis.org

53, 239, 6719, 7321, 4073561, 6257813, 6521843, 85271063

Offset: 1

Zak Seidov, Mar 25 2015

The last term is a(8) = 85271063 (prime) because all 8 digits are different and m=1063 with 8527=prime(m).

Subsequence of A029743 (distinct-digit primes).

Cf. A073643, A074665, A074667, A074669, A074671, A074673, A074675, A160402.

Select[FromDigits[IntegerDigits[Prime@ #]~Join~IntegerDigits@ #] & /@
Range@ 1200, PrimeQ@ # && Max@ DigitCount@ # == 1 &] (* Michael De Vlieger, Mar 25 2015 *)

A259146 Smallest prime with first n digits distinct.

Original entry on oeis.org

2, 13, 103, 1039, 10243, 102359, 1023467, 10234589, 102345689, 10234567897

Offset: 1

Zak Seidov, Jun 19 2015

The sequence is complete: n=1..10.

Cf. A000040, A007809, A007810, A029743.

from sympy import nextprime
def a(n):
  p = nextprime(10**(n-1))
  while len(set(str(p)[:n])) < n: p = nextprime(p)
  return p
for n in range(1, 11):
  print(a(n), end=", ") # Michael S. Branicky, Feb 13 2021

a(n) = A007809(n), n<=9. - R. J. Mathar, Jul 06 2015

A259152 a(n) = smallest n-digit prime with first 10 digits distinct.

Original entry on oeis.org

10234567897, 102345678907, 1023456789013, 10234567890077, 102345678900007, 1023456789000073, 10234567890000053, 102345678900000059, 1023456789000000049, 10234567890000000007

Offset: 11

Zak Seidov, Jun 19 2015

There is no 10-digit prime with the first 10 digits distinct, hence offset=11.

Robert Israel, Table of n, a(n) for n = 11..900

Cf. A000040, A007810, A029743, A259146.

seq(nextprime(1023456789*10^(d-10)),d=11..100); # Robert Israel, Jun 19 2015

Table[NextPrime[1023456789*10^(d - 10)], {d, 11, 100}] (* Michael De Vlieger, Jun 19 2015, after the Maple by Robert Israel *)

use Math::GMPz; use ntheory ":all"; do { my $n=next_prime(1023456789 * Math::GMPz->new(10)**($-10)); say $n; } for (11..100); # _Dana Jacobsen, Jun 26 2015

A259187 Primes p such that both p and p^2 are distinct-digit numbers.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 43, 53, 59, 61, 71, 73, 79, 89, 137, 179, 193, 269, 281, 367, 397, 463, 487, 509, 571, 593, 647, 709, 829, 839, 1307, 1873, 2069, 2731, 2801, 3041, 4157, 4967, 4987, 6043, 7549, 7621, 8623, 21397

Offset: 1

Zak Seidov, Jun 20 2015

Corresponding squares are 4, 9, 25, 49, 169, 289, 361, 529, 841, 961, 1369, 1849, 2809, 3481, 3721, 5041, 5329, 6241, 7921, 18769, 32041, 37249, 72361, 78961, 134689, 157609, 214369, 237169, 259081, 326041, 351649, 418609, 502681, 687241, 703921, 1708249, 3508129, 4280761, 7458361, 7845601, 9247681, 17280649, 24671089, 24870169, 36517849, 56987401, 58079641, 74356129, 457831609 (subsequence of A078255).

Subsequence of A029743 and of A119509. Cf. A078255.

Select[Prime[Range[2500]],Max[DigitCount[#]]<2&&Max[DigitCount[#^2]]<2&] (* Harvey P. Dale, May 25 2020 *)

A262250 Primes having exactly one digit in {1, 3, 7, 9}.

Original entry on oeis.org

3, 7, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 223, 227, 229, 241, 251, 257, 263, 269, 281, 283, 401, 409, 421, 443, 449, 457, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 557, 563, 569, 587, 601, 607, 641, 643, 647, 653, 659, 661, 683, 809, 821, 823, 827, 829, 853, 857, 859, 863, 881, 883, 887, 2003, 2027, 2029, 2053

Offset: 1

Vladimir Shevelev, Sep 21 2015

How can one prove that the sequence is infinite?

Probabilistic arguments imply that the number of terms not exceeding x is not less than 5/9*log(10)/log(6)*x^(log(6)/log(10))/log(x) = 0.7139...*x^0.778.../log(x).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A029743.

Select[Prime@ Range@ 310, Total@ Drop[First /@ Partition[DigitCount@ #, 2, 2], {3}] == 1 &] (* Michael De Vlieger, Sep 21 2015 *)

nbd(vd, d) = #select(i->(i == d), vd);
lista(nn) = {forprime(p=2, nn, vd = digits(p); if (nbd(vd,1) + nbd(vd,3) + nbd(vd,7) + nbd(vd,9) == 1, print1(p, ", ")););} \\ Michel Marcus, Sep 22 2015

list(lim)=my(v=List([3])); forprime(p=7,lim, if(#setintersect(Set(digits(p\10)), [1,3,7,9])==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Sep 22 2015

use ntheory ":all"; say join ", ", grep { tr/1379// == 1 } @{primes(3000)}; # Dana Jacobsen, Oct 13 2015

A323391 Primes containing nonprime digits (from 1 to 9) in their decimal expansion and whose digits are distinct, i.e., consisting of only digits 1, 4, 6, 8, 9.

Original entry on oeis.org

19, 41, 61, 89, 149, 419, 461, 491, 619, 641, 691, 941, 1489, 4691, 4861, 6481, 6491, 6841, 8419, 8461, 8641, 8941, 9461, 14869, 46819, 48619, 49681, 64189, 64891, 68491, 69481, 81649, 84691, 84961, 86491, 98641

Offset: 1

Bernard Schott, Jan 13 2019

There are only 36 terms in this sequence, which is a finite subsequence of A152313.
Two particular examples:
6481 is also the smallest prime formed from the concatenation of two consecutive squares.
81649 is the only prime containing all the nonprime positive digits such that every string of two consecutive digits is a square.

			14869 is the smallest prime that contains all the nonprime positive digits; 98641 is the largest one.

Chris K. Caldwell and G. L. Honaker, Jr., 81649, Prime Curios!

Subsequence of A152313. Subsequence of A029743. Subsequence of A155024 (with distinct nonprime digits but with 0) and of A034844.

Cf. A029743 (with distinct digits), A124674 (with distinct prime digits), A155045 (with distinct odd digits), A323387 (with distinct square digits), A323578 (with distinct digits for which parity of digits alternates).

Select[Union@ Flatten@ Map[FromDigits /@ Permutations@ # &, Rest@ Subsets@ {1, 4, 6, 8, 9}], PrimeQ] (* Michael De Vlieger, Jan 19 2019 *)

isok(p) = isprime(p) && (d=digits(p)) && vecmin(d) && (#Set(d) == #d) && (#select(x->isprime(x), d) == 0); \\ Michel Marcus, Jan 14 2019

A385711 Primes whose digits are all distinct and pairwise coprime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 137, 149, 157, 167, 173, 179, 197, 251, 257, 271, 317, 347, 419, 431, 457, 479, 491, 521, 523, 541, 547, 571, 587, 617, 719, 743, 751, 761, 853, 857, 859, 941, 947, 971, 1237, 1259

Offset: 1

Gonzalo Martínez, Jul 07 2025

This sequence has 252 terms, the last being 95471, which have at most 5 digits. This is because each term has at most one even digit and at most 4 odd digits, since gcd(3, 9) = 3.

All terms are in A038618, since if zero is among the digits of a prime p, then p must have at least 3 digits, where at least one of them is greater than 1, say d, and in such a case gcd(0, d) = d ! = 1.

			857 is a term since it is prime and gcd(8, 5) = gcd(5, 7) = gcd(8, 7) = 1.

Gonzalo Martínez, Table of n, a(n) for n = 1..252

Subsequence of A029743.

Cf. A038618.

Select[Prime[Range[10000]], UnsameQ @@ (d = IntegerDigits[#]) && AllTrue[Subsets[d, {2}], CoprimeQ @@ # &] &] (* Amiram Eldar, Jul 13 2025 *)

A085453 Numbers n such that n^2 and n^3 together use only distinct digits.

Original entry on oeis.org

2, 3, 8, 9, 24, 69

Offset: 1

Zak Seidov, Jul 01 2003

There are only six such numbers (in base 10). Numbers with distinct digits in A010784. Primes with distinct digits in A029743. The case n and n^2 (exactly 22 numbers) in A059930. The case n and prime[n] (exactly 101 numbers) in A085451.

			69 is (the last) term because 69^2=4761 and 69^3=328509 together use all 10 distinct digits.

Cf. A010784 A029743 A059930 A085451

bb = {}; Do[idpn = IntegerDigits[n^3]; idn = IntegerDigits[n^2]; If[Length[idn] + Length[idpn] == Length[Union[idn, idpn]], bb = {bb, n}], {n, 1, 10000}]; Flatten[bb]

A250173 Primes with distinct digits: a(n) is the least prime > a(n-1) such that a(n-1) and a(n) share no common digit.

Original entry on oeis.org

2, 5, 7, 13, 29, 31, 47, 53, 61, 73, 89, 103, 257, 349, 521, 607, 821, 907, 1283, 4057, 6329, 7451, 8039, 12457, 30689, 41257, 63809, 74521, 80369

Offset: 1

Zak Seidov, Dec 05 2014

Subsequence of A029743.

Cf. A030284, A249953.

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A132129 Largest prime with distinct digits when written in base n.

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A256339 Distinct-digit primes that are concatenation of prime(m) and m for some m.

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A259146 Smallest prime with first n digits distinct.

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A259152 a(n) = smallest n-digit prime with first 10 digits distinct.

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A259187 Primes p such that both p and p^2 are distinct-digit numbers.

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A262250 Primes having exactly one digit in {1, 3, 7, 9}.

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A323391 Primes containing nonprime digits (from 1 to 9) in their decimal expansion and whose digits are distinct, i.e., consisting of only digits 1, 4, 6, 8, 9.

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A385711 Primes whose digits are all distinct and pairwise coprime.

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