A007810 -id:A007810 - cp's OEIS frontend

A100369 Largest primes arising in A099756 which were built up from n distinct digits. This sequence differs from A007810 because more than one copy of each digit is permitted.

11, 787, 22259, 70879, 607889, 4456789, 40456789, 304456879, 1123465789, 10123457689

Offset: 1

Labos Elemer, Nov 29 2004

These primes are "largest among earliest primes" at fixed number of distinct digits chosen from A099756. Their position in A099756 are: {1,27,90,198,440,774,858,930,949,950}.

			n=3: a[3]=22259, built up from the 3-subset = {2,5,9} of decimal digits.
It appears in A099756 as the 90th term. It is by definition of A099756 is
the smallest prime that can be constructed from {2,5,9} and at the same time
it is the largest prime if running through all 3-subsets of decimal digits.
All terms includes at least 2 copies of some digit.
Differs from A007810[3]=983.

Cf. A099651, A099653, A099654, A007809, A007810, A099756.

Edited by Charles R Greathouse IV, Aug 03 2010

A029743 Primes with distinct digits.

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, 103, 107, 109, 127, 137, 139, 149, 157, 163, 167, 173, 179, 193, 197, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 307, 317, 347, 349, 359, 367, 379, 389

Offset: 1

Patrick De Geest

This sequence has 283086 terms, the last being 987654103 = A007810(9). - Jud McCranie

Intersection of A010784 and A000040; A178788(a(n)) * A010051(a(n)) = 1. [Reinhard Zumkeller, Sep 25 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..283086, full sequence
Patrick De Geest, World!Of Numbers

Cf. A000040, A007810, A010051, A010784, A178788.

a029743 n = a029743_list !! (n-1)
a029743_list = filter ((== 1) . a010051) a010784_list
-- Reinhard Zumkeller, Sep 25 2011

t={};Do[p=Prime[n];If[Select[Transpose[Tally[IntegerDigits[p]]][[2]],#>1 &]=={},AppendTo[t,p]],{n,77}];t (* Jayanta Basu, May 04 2013 *)
Select[Prime[Range[80]],Max[DigitCount[#]]<2&] (* Harvey P. Dale, Sep 13 2020 *)

from sympy import isprime
from itertools import permutations as P
dist = [p for d in range(1, 11) for p in P("0123456789", d) if p[0] != "0"]
afull = [t for t in (int("".join(p)) for p in dist) if isprime(t)]
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

A007809 Smallest prime with n distinct digits.

Original entry on oeis.org

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

Offset: 1

N.B. Backhouse (sx52(AT)liverpool.ac.uk)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1039, p.126, Ellipses, Paris 2008. [From Lekraj Beedassy, Oct 12 2008]

Cf. A038378, A007810, A071360, A071361, A071362, A071363.

Table[Module[{k=NextPrime[10^n]},While[Max[DigitCount[k]]>1,k=NextPrime[k]];k],{n,0,8}] (* Harvey P. Dale, May 27 2025 *)

A007809(n,p=A038378(n))={until(isprime(p),while(#Set(digits(p++))M. F. Hasler, May 04 2017

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

Corrected by Jud McCranie, Jan 03 2001

A071361 Largest n-digit prime with strictly decreasing digits.

Original entry on oeis.org

7, 97, 983, 9871, 98731, 987631, 9875321, 98765431

Offset: 1

Rick L. Shepherd, May 21 2002

Differs from A007810 (largest prime with n distinct digits) from a(7) on. [Edited by M. F. Hasler, May 03 2017]

Cf. A071360, A071362, A071363, A007810.

A071361(n,u=-vectorv(n,i,10^(n-i)))=forvec(d=vector(n,i,-[9,1]),isprime(d*u)&&return(d*u),2) \\ M. F. Hasler, May 03 2017

A071363 Largest n-digit prime with strictly increasing digits.

Original entry on oeis.org

7, 89, 569, 5689, 34679, 345689, 1456789, 23456789

Offset: 1

Rick L. Shepherd, May 21 2002

Notice the terms with consecutive digits; search for 23456789 to find several related sequences including A006055, A052017 and A052077.

			a(1) = A052015(4), a(2) = A052015(15), a(3) = A052015(35), a(4) = A052015(61), ... In short, a(n) = A052015(b(n)) with b = (4, 15, 35, 61, 81, 94, 98, 100). - _M. F. Hasler_, May 03 2017

Cf. A071362, A071360, A071361, A007810.

Subsequence of A052015.

A071363(n,u=vectorv(n,i,10^(n-i)))={forvec(d=vector(n,i,[1,9]),isprime(d*u)&&n=d*u,2);n} \\ M. F. Hasler, May 03 2017

A100370 Primes in A099756, sorted.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 79, 83, 89, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 179, 181, 211, 227, 239, 241, 251, 257, 263, 269, 281, 283, 307, 347, 349, 359, 367, 379, 389, 401, 409, 431, 449, 457, 461

Offset: 1

Labos Elemer, Nov 30 2004

Inspired by A099756.

			Positions of "minimal terms" (see A007809) inside A099756 and here, in A100370, are {2,8,31,138,320,574,779,900,942,950} or {1,6,23,84,250,494,721,873,934,950} respectively.
This is because the orders of A099756 and A100370 are based on different criteria.

Robert G. Wilson v, Table of n, a(n) for n = 1..950

Cf. A007809, A007810, A099651, A099653, A099654, A099756, A100369.

< 0 &][[-2]]*10^(Length[ss[[n]]] -  If[ Mod[ FromDigits@ ss[[n]], 3] == 0, 0, 1]) - 1]}, While[ Union@ IntegerDigits@ p != id, p = NextPrime@ p]; p]; f[3] = 3; Sort@ Array[f, 950]

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

Original entry on oeis.org

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

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

A158716 Largest prime with n distinct nonprime digits.

Original entry on oeis.org

89, 941, 9601, 98641, 980641

Offset: 2

Lekraj Beedassy, Mar 24 2009

Cf. A007810.

a[n_]:=Module[{k=NextPrime[10^n-1,-1]}, While[!SubsetQ[{0,1,4,6,8,9}, IntegerDigits[k]] || CountDistinct[IntegerDigits[k]]Stefano Spezia, Aug 29 2025 *)

cp's OEIS Frontend

A100369 Largest primes arising in A099756 which were built up from n distinct digits. This sequence differs from A007810 because more than one copy of each digit is permitted.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A029743 Primes with distinct digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A007809 Smallest prime with n distinct digits.

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A071361 Largest n-digit prime with strictly decreasing digits.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A071363 Largest n-digit prime with strictly increasing digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A100370 Primes in A099756, sorted.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Extensions

A259146 Smallest prime with first n digits distinct.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

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

Views