A010784 -id:A010784 - cp's OEIS frontend

A305714 Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.

1, 1, 1, 2, 0, 0, 2, 0, 1, 1, 1

Offset: 0

Gus Wiseman, Jun 08 2018

A sequence q of length k is strictly pandigital if it is a permutation of {1,2,...,k}. It is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

			Sequence of sets of n-digit numbers that are weakly polydivisible and strictly pandigital is (with A = 10):
  {0}
  {1}
  {12}
  {123,321}
  {}
  {}
  {123654,321654}
  {}
  {38165472}
  {381654729}
  {381654729A}

Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305715.

A321682 Numbers with distinct digits in factorial base.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 13, 14, 19, 20, 22, 23, 46, 67, 68, 77, 82, 85, 86, 101, 106, 109, 110, 115, 116, 118, 119, 238, 355, 356, 461, 466, 469, 470, 503, 526, 547, 548, 557, 562, 565, 566, 623, 646, 667, 668, 677, 682, 685, 686, 701, 706, 709, 710, 715, 716, 718

Offset: 1

Rémy Sigrist, Nov 16 2018

This sequence is a variant of A010784; however here we have infinitely many terms (for example all the terms of A033312 belong to this sequence).

			The first terms, alongside the corresponding factorial base representations, are:
  n   a(n)  fac(a(n))
  --  ----  ---------
   1     0        (0)
   2     1        (1)
   3     2      (1,0)
   4     4      (2,0)
   5     5      (2,1)
   6    10    (1,2,0)
   7    13    (2,0,1)
   8    14    (2,1,0)
   9    19    (3,0,1)
  10    20    (3,1,0)
  11    22    (3,2,0)
  12    23    (3,2,1)
  13    46  (1,3,2,0)
  14    67  (2,3,0,1)

Rémy Sigrist, Table of n, a(n) for n = 1..8179 (terms up to 13!)
Index entries for sequences related to factorial base representation.

Cf. A010784, A033312, A108731.

b:= proc(n, i) local r; `if`(n (l-> is(nops(l)=nops({l[]})))(b(n, 2)):
select(t, [$0..1000])[];  # Alois P. Heinz, Nov 16 2018

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; UnsameQ @@ s]; Select[Range[0, 720], q] (* Amiram Eldar, Feb 21 2024 *)

is(n) = my (s=0); for (k=2, oo, if (n==0, return (1)); my (d=n%k); if (bittest(s,d), return (0), s+=2^d; n\=k))

A342383 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that both the digits in a(n) and the digits in a(n-1)+a(n) are all distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 13, 12, 14, 15, 16, 18, 17, 19, 20, 21, 24, 23, 25, 26, 27, 29, 28, 30, 31, 32, 35, 34, 36, 37, 38, 40, 39, 41, 42, 43, 46, 45, 47, 48, 49, 53, 50, 52, 51, 54, 69, 56, 64, 59, 61, 62, 58, 65, 60, 63, 57, 67, 68, 70, 72, 71, 74, 73, 75, 78, 76, 80, 79, 81, 82, 83

Offset: 0

Scott R. Shannon, Mar 09 2021

The sequence is finite due to the finite number of positive integers with distinct digits, see A010784, although the exact number of terms is currently unknown.

			a(1) = 1 as 1 has one distinct digit and a(0)+1 = 0+1 = 1 which has one distinct digit 0.
a(6) = 7 as 7 has one distinct digit and a(5)+7 = 5+7 = 12 which has two distinct digits. Note that 6 is the first skipped number as a(5)+6 = 5+6 = 11 has 1 as a duplicate digit.
a(11) = 13 as 13 has two distinct digits and a(10)+13 = 10+13 = 23 which has two distinct digits. Note that 11 and 12 are skipped as 11 has 1 as a duplicate digit while a(10)+12 = 10+12 = 22 has 2 as a duplicate digit.

Scott R. Shannon, Image of the first 60000 terms. The green line is a(n) = n.

Cf. A336285, A342382, A010784, A043537, A043096, A276633, A002378.

Block[{a = {0}, k, m = 10^4}, Do[k = 1; While[Nand[FreeQ[a, k], AllTrue[DigitCount[a[[-1]] + k], # < 2 &], AllTrue[DigitCount[k], # < 2 &]], If[k > m, Break[]]; k++]; If[k > m, Break[]]; AppendTo[a, k], {i, 76}]; a] (* Michael De Vlieger, Mar 11 2021 *)

def agen():
  alst, aset = [0], {0}
  yield 0
  while True:
    an = 1
    while True:
      while an in aset: an += 1
      stran, t = str(an), str(alst[-1] + an)
      if len(stran) == len(set(stran)) and len(t) == len(set(t)):
        alst.append(an); aset.add(an); yield an; break
      an += 1
g = agen()
print([next(g) for n in range(77)]) # Michael S. Branicky, Mar 11 2021

A343921 The maximum number of times a positive number can be added to n such that the digits in each resulting sum are distinct.

Original entry on oeis.org

36, 9, 12, 13, 12, 11, 15, 12, 11, 26, 14, 13, 23, 11, 11, 13, 26, 11, 12, 12, 13, 23, 14, 11, 24, 12, 13, 35, 25, 12, 12, 16, 13, 12, 12, 11, 13, 11, 17, 12, 13, 12, 15, 9, 12, 12, 25, 9, 14, 22, 12, 23, 12, 25, 34, 11, 11, 13, 22, 11, 16, 12, 14, 12, 12, 24, 13, 13, 15, 12, 13, 10, 11, 11, 9

Offset: 0

Scott R. Shannon, May 04 2021

See A338659 for the smallest positive number that can be added to n a total of a(n) times such that the digits in each resulting sum are distinct.

See A343922 for the largest positive number that can be added to n a total of a(n) times such that the digits in each resulting sum are distinct.

			a(8) = 11 as A338659(8) = A343922(8) = 150 can be added to 8 a total of 11 times with each sum containing distinct digits. The sums are 158, 308, 458, 608, 758, 908, 1058, 1208, 1358, 1508, 1658. No other positive number can be added to 8 a total of 11 or more times to produce such sums.

Cf. A338659, A343922, A010784, A043096, A029743.

a(n) = 0 for n >= 9876543210.

A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 130, 131, 132, 140, 141, 142, 143, 150

Offset: 1

Bernard Schott, Jun 27 2022

This definition comes from Patrick De Geest's link.
Other definitions for undulating are present in the OEIS (e.g., A033619, A046075).
When the absolute differences between two adjacent digits are always equal (e.g., 85858), these numbers are called smoothly undulating numbers and form a subsequence (A046075).
The definition includes the trivial 1- and 2-digit undulating numbers.
Subsequence of A043096 where the first different term is A043096(103) = 123 while a(103) = 130.
This sequence first differs from A010784 at a(92) = 101, A010784(92) = 102.
The sequence differs from A160542 (which contains 100). - R. J. Mathar, Aug 05 2022

			111 is not a term here, but A033619(102) = 111.
a(93) = 102, but 102 is not a term of A046075.
Some terms: 5276, 918230, 1053837, 263915847, 3636363636363636.
Are not terms: 1331, 594571652, 824327182.

Patrick De Geest, Smoothly Undulating Palindromic Primes, World of Numbers.

Cf. A033619, A043096, A046075.
Cf. A059168 (subsequence of primes).
Differs from A010784, A241157, A241158.
Cf. A355302, A355303, A355304.

isA355301 := proc(n)
    local dgs,i,back,forw ;
    dgs := convert(n,base,10) ;
    if nops(dgs) < 2 then
        return true;
    end if;
    for i from 2 to nops(dgs)-1 do
        back := op(i,dgs) -op(i-1,dgs) ;
        forw := op(i+1,dgs) -op(i,dgs) ;
        if back*forw >= 0 then
            return false;
        end if ;
    end do:
    back := op(-1,dgs) -op(-2,dgs) ;
    if back = 0 then
        return false;
    end if ;
    return true ;
end proc:
A355301 := proc(n)
    option remember ;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA355301(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A355301(n),n=1..110) ; # R. J. Mathar, Aug 05 2022

q[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; Select[Range[0, 100], q] (* Amiram Eldar, Jun 28 2022 *)

isok(m) = if (m<10, return(1)); my(d=digits(m), dd = vector(#d-1, k, sign(d[k+1]-d[k]))); if (#select(x->(x==0), dd), return(0)); my(pdd = vector(#dd-1, k, dd[k+1]*dd[k])); #select(x->(x>0), pdd) == 0; \\ Michel Marcus, Jun 30 2022

A029741 Even numbers with distinct digits.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 90, 92, 94, 96, 98, 102, 104, 106, 108, 120, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 146

Offset: 1

Patrick De Geest

Largest term is 9876543210. - Alonso del Arte, Jan 09 2020

There are 4493646 terms. - Michael S. Branicky, Aug 04 2022

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers

Cf. A029740 (odd version). Union of that sequence with this sequence gives A010784.

Select[2Range[0, 79], Max[DigitCount[#]] == 1 &] (* Harvey P. Dale, Dec 23 2013 *)

# generates full sequence
from itertools import permutations
afull = [0] + sorted(set(int("".join(p)) for d in range(1, 11) for p in permutations("0123456789", d) if p[0] != "0" and p[-1] in "02468"))
print(afull[:100]) # Michael S. Branicky, Aug 04 2022

def hasDistinctDigits(n: Int): Boolean = {
  val numerStr = n.toString
  val digitSet = numerStr.split("").toSet
  numerStr.length == digitSet.size
}
(0 to 198 by 2).filter(hasDistinctDigits) // Alonso del Arte, Jan 09 2020

Offset changed to 1 by Michael S. Branicky, Aug 04 2022

A085451 Numbers n such that n and prime[n] together use only distinct digits.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 35, 39, 40, 45, 53, 57, 58, 60, 61, 69, 70, 72, 79, 85, 89, 90, 91, 93, 96, 98, 104, 108, 120, 124, 145, 146, 147, 150, 162, 236, 237, 253, 254, 259, 315, 316, 359, 380, 384, 390, 405, 406, 460, 461, 518

Offset: 1

Zak Seidov, Jul 01 2003

There are exactly 101 such numbers in the sequence. Numbers with distinct digits in A010784. Primes with distinct digits in A029743. The case n and n^2 (exactly 22 numbers) in A059930.

A178788(A045532(a(n))) = 1. [From Reinhard Zumkeller, Jun 30 2010]

			3106 is in the sequence (and the last term) because it and prime[3106]=28549 together use all 10 distinct digits.

Giovanni Resta, Table of n, a(n) for n = 1..101 (full sequence)

Cf. A010784 A029743 A059930 A085452.

bb = {}; Do[idpn = IntegerDigits[Prime[n]]; idn = IntegerDigits[n]; If[Length[idn] + Length[idpn] == Length[Union[idn, idpn]], bb = {bb, n}], {n, 1, 100000}]; Flatten[bb]

A107846 Number of duplicate digits of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0

Offset: 0

Rick L. Shepherd, May 24 2005

a(A010784(n)) = 0; a(A109303(n)) > 0. - Reinhard Zumkeller, Jul 09 2013

			a(11) = 1 because 11 has two total decimal digits but only one distinct digit (1) and 2-1=1.
Similarly, a(3653135) = 7 (total digits) - 4 (distinct digits: 1,3,5,6) = 3 (There are three duplicate digits here, namely, 3, 3 and 5).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A055642 (Total decimal digits of n), A043537 (Distinct decimal digits of n).

import Data.List (sort, group)
a107846 = length . concatMap tail . group . sort . show :: Integer -> Int
-- Reinhard Zumkeller, Jul 09 2013

Table[Total[Select[DigitCount[n]-1,#>0&]],{n,0,120}] (* Harvey P. Dale, Jul 31 2013 *)

def a(n): return len(s:=str(n)) - len(set(s))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jan 09 2023

a(n) = A055642(n) - A043537(n).

A241157 Numbers in which the two least-significant digits are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 1

N. J. A. Sloane, Apr 18 2014

More than the usual number of terms are displayed in order to show the difference from some closely-related sequences.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010784, A241158.

a241157 n = a241157_list !! (n-1)
a241157_list = 0 : filter f [0..] where
   f x = d' /= d where d' = mod x' 10; (x', d) = divMod x 10
-- Reinhard Zumkeller, May 02 2014

A241158 Numbers in which the two leading (most significant) digits are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120

Offset: 1

N. J. A. Sloane, Apr 18 2014

More than the usual number of terms are displayed in order to show the difference from some closely-related sequences.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010784, A043096, A241157.

a241158 n = a241158_list !! (n-1)
a241158_list = filter (f . show)  [0..] where
   f [] = True; f (d : d' : ) = d /= d'
-- Reinhard Zumkeller, May 02 2014

cp's OEIS Frontend

A305714 Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.

Views

Author

Keywords

Comments

Examples

Crossrefs

A321682 Numbers with distinct digits in factorial base.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A342383 a(0) = 0; for n > 0, a(n) is the least positive integer not occurring earlier such that both the digits in a(n) and the digits in a(n-1)+a(n) are all distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A343921 The maximum number of times a positive number can be added to n such that the digits in each resulting sum are distinct.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A029741 Even numbers with distinct digits.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Scala

Extensions

A085451 Numbers n such that n and prime[n] together use only distinct digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A107846 Number of duplicate digits of n.

Views

Author

Keywords

Comments

Examples

Links