A007809 -id:A007809 - cp's OEIS frontend

A071362 Smallest n-digit prime with strictly increasing digits.

2, 13, 127, 1237, 12347, 123457, 1234789, 12356789

Offset: 1

Rick L. Shepherd, May 21 2002

A071362(n, u=vectorv(n, i, 10^(n-i)))=forvec(d=vector(n, i, [1, 9]), isprime(d*u)&&return(d*u), 2) \\ M. F. Hasler, May 04 2017

from sympy import nextprime
def inc(n):
  s = str(n); return len(s)==1 or all(d>pd for pd, d in zip(s[:-1], s[1:]))
def a(n):
  p = nextprime(10**(n-1))
  while not inc(p): p = nextprime(p)
  return p
for n in range(1, 9):
  print(a(n), end=", ") # Michael S. Branicky, Feb 13 2021

A038378 Integers which have more distinct digits than any smaller number.

Original entry on oeis.org

0, 10, 102, 1023, 10234, 102345, 1023456, 10234567, 102345678, 1023456789

Offset: 1

N. J. A. Sloane

Or: Smallest number with exactly n distinct digits. - M. F. Hasler, May 04 2017

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A007809, A043537, A086069.

Prepend[NestList[FromDigits[Append[IntegerDigits[#],Last[Union[IntegerDigits[#]]]+1]]&,10,8],0] (* Ivan N. Ianakiev, May 10 2015 *)
Join[{0},Table[FromDigits[PadRight[{1,0},n,{1,0,2,3,4,5,6,7,8,9}]],{n,2,10}]] (* Harvey P. Dale, Sep 27 2016 *)

A038378(n)=sum(i=1,n--,10^(n-i)*if(i>1,i,10)) \\ M. F. Hasler, May 04 2017

Offset fixed by Reinhard Zumkeller, Aug 25 2009

A071360 Smallest n-digit prime with strictly decreasing digits.

Original entry on oeis.org

2, 31, 421, 5431, 75431, 764321, 8764321, 97654321

Offset: 1

Rick L. Shepherd, May 21 2002

Cf. A071361, A071362, A071363, A007809.

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.

Original entry on oeis.org

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

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]

A141116 Smallest n-digit prime with no identical adjacent digits (or 0 if no such prime exists).

Original entry on oeis.org

2, 13, 101, 1013, 10103, 101021, 1010129, 10101023, 101010157, 1010101039, 10101010163, 101010101063, 1010101010131, 10101010101019, 101010101010131, 1010101010101037, 10101010101010141, 101010101010101083

Offset: 1

Rick L. Shepherd, Jun 05 2008

For n >= 1, a(n) >= A056830(n), the least n-digit positive integer with no identical adjacent digits (also the least positive integer whose digits occur in n runs). Conjecture: For all n, a(n) <> 0.
If the conjecture is true, then this sequence and the following two sequences are equivalent: i) Smallest prime with exactly n runs of digits and ii) Smallest prime with at least n runs of digits. For each n <= 625, a(n) is an n-digit prime (provided that each probable prime shown in the link is indeed a prime -- or at least one of very many (slightly) larger probable prime candidates is prime).
As each a(n) shown is very near A056830(n), I believe it is extremely unlikely that a randomly-given n would yield a 0 term (but I don't have a proof for arbitrary n).

			a(4) = 1013 because 1013 is the smallest 4-digit prime having no identical adjacent digits; the only smaller 4-digit prime, 1009, is disqualified by the "00", identical adjacent digits (of run length 2). Also each digit, 1, 0, 1, 3, occurs in a run of identical digits of length 1 for a total of 4 runs with 1013 being the smallest prime of any length with 4 runs of digits.

Rick L. Shepherd, Table of n, a(n) for n = 1..625

Cf. A003617, A007809, A056830.

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

A360505 Concatenate the ternary strings for n, n-1, n-2, ..., 2, 1.

Original entry on oeis.org

1, 21, 1021, 111021, 12111021, 2012111021, 212012111021, 22212012111021, 10022212012111021, 10110022212012111021, 10210110022212012111021, 11010210110022212012111021, 11111010210110022212012111021, 11211111010210110022212012111021, 12011211111010210110022212012111021

Offset: 1

N. J. A. Sloane, Feb 17 2023

Similar to A360502, but here we count down from n to 1 in base 3 and concatenate the strings to get a(n).

This is the ternary analog of A000422.

Index entries for sequences related to Most Wanted Primes video

Cf. A000422, A007809, A360502, A360506.

a[n_] := FromDigits @ Flatten @ IntegerDigits[Range[n, 1, -1], 3]; Array[a, 15] (* Amiram Eldar, Feb 18 2023 *)

a(n) = strjoin(concat([digits(k, 3) | k <- Vecrev([1..n])])) \\ Rémy Sigrist, Feb 18 2023

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(n, 0, -1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:="".join(map(str, digits(n, 3)[1:]))+s) for n in count(1))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

A081716 Smallest number x such that phi(x)=A000010(x) has exactly n different decimal digits.

Original entry on oeis.org

2, 11, 103, 1033, 10247, 102359, 1023487, 10234759, 102345979, 1023475969

Offset: 1

Labos Elemer, Apr 04 2003

			phi applied to the sequence gives: 1, 10, 102, 1032, 10246, 102358, 1023486, 10234758, 102345978, ...

Cf. A007809, A038378, A000010.

f[x_] := Length[Union[IntegerDigits[x]]]
t=Table[0, {10}];
Do[s=f[n]; If[PrimeQ[n]&&s<11&&t[[s]]==0,
t[[s]]=n], {n, 1, 10000000000}];

a(10) from Donovan Johnson, Feb 05 2010

A141405 Smallest n-digit prime using the most distinct and consecutive digits.

Original entry on oeis.org

2, 23, 101, 1423, 10243, 102043, 1234657, 10243567, 100234657, 1123465789, 10123457689, 100123456789, 1001233458679, 10011223456879, 100122334546789, 1001122334546879

Offset: 1

Lekraj Beedassy, Aug 03 2008, Aug 09 2008

			a(10)=1123465789 and not 1002430567 because although the latter prime is smaller it does not have the most distinct digits. Similarly,a(13)= 1001233458679 and not 1001123456897. Note that for n >=11,all digits 0 through 9 are used.

Cf. A007809 [From Lekraj Beedassy, Sep 14 2008]

cp's OEIS Frontend

A071362 Smallest n-digit prime with strictly increasing digits.

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A038378 Integers which have more distinct digits than any smaller number.

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A071360 Smallest n-digit prime with strictly decreasing digits.

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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.

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A100370 Primes in A099756, sorted.

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A141116 Smallest n-digit prime with no identical adjacent digits (or 0 if no such prime exists).

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A259146 Smallest prime with first n digits distinct.

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A360505 Concatenate the ternary strings for n, n-1, n-2, ..., 2, 1.

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A081716 Smallest number x such that phi(x)=A000010(x) has exactly n different decimal digits.

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