A171102 -id:A171102 - cp's OEIS frontend

A047726 Number of different numbers that are formed by permuting digits of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 6

Offset: 1

N. J. A. Sloane

The minimum value of a(A171102(n)) is 10*9!. - Altug Alkan, Jul 08 2016

			From 102 we get 102, 120, 210, 201, 12 and 21, so a(102)=6.
From 33950 with 5 digits, one '0', two '3', one '5' and one '9', we get 5! / (1! * 2! * 1! * 1!) = 60 different numbers and a(33950) = 60.  - _Bernard Schott_, Oct 20 2019

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.

Cf. A055098. Identical to A043537 and A043562 for n<100.

Cf. A179239. - Aaron Dunigan AtLee, Jul 14 2010

import Data.List (permutations, nub)
a047726 n = length $ nub $ permutations $ show n
-- Reinhard Zumkeller, Jul 26 2011

f:= proc(n) local L;
  L:= convert(n,base,10);
  nops(L)!/mul(numboccur(i,L)!,i=0..9);
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 08 2016

pd[n_]:=Module[{p=Permutations[IntegerDigits[n]]},Length[Union [FromDigits/@p]]]; pd/@Range[120]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=n=eval(Vec(Str(n)));(#n)!/prod(i=0,9,sum(j=1,#n,n[j]==i)!) \\ Charles R Greathouse IV, Sep 29 2011

A047726(n)={local(c=Vec(0,10)); apply(d->c[d+1]++, digits(n)); logint(n*10,10)!/prod(i=1,10,c[i]!)} \\ M. F. Hasler, Oct 18 2019

a(n) << n / (log_10 n)^4.5 by Stirling's approximation. - Charles R Greathouse IV, Sep 29 2011

a(n) = A000142(A055642(n))/Product_{k=0..9} A000142(A100910(n,k)). - Robert Israel, Jul 08 2016

Corrected by Henry Bottomley, Apr 19 2000

A029793 Numbers k such that k and k^2 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215, 47620, 48302, 48320, 49827, 64510, 68474, 71205, 72510, 72576, 74510, 74528, 79286, 79603, 79836, 94583, 94867, 96123, 98376

Offset: 1

Patrick De Geest

This sequence has density 1: almost all numbers k have all 10 digits in both k and k^2. - Franklin T. Adams-Watters, Jun 28 2011

			{0, 1, 3, 4, 9} = digits of a(10) = 10493 and of 10493^2 = 110103049;
{0, 1, 2, 5, 6} = digits of a(100) = 162025 and of 162025^2 = 26252100625;
{0, 1, 3, 4, 6, 7, 8} = digits of a(1000) = 1764380 and of 1764380^2 = 3113036784400;
{1, 2, 3, 4, 7, 8, 9} = digits of a(10000) = 14872239 and of 14872239^2 = 221183492873121.

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

Cf. A029795, A029797, A030091; A178501 is a subsequence, A054038, A171102, A232659-A232662.

import Data.List (nub, sort)
a029793 n = a029793_list !! (n-1)
a029793_list = filter (\x -> digs x == digs (x^2)) [0..]
   where digs = sort . nub . show
-- Reinhard Zumkeller, Jun 27 2011

[ n: n in [0..10^5] | Set(Intseq(n)) eq Set(Intseq(n^2)) ];  // Bruno Berselli, Jun 28 2011

seq(`if`(convert(convert(n,base,10),set) = convert(convert(n^2,base,10),set), n, NULL), n=0..100000); # Nathaniel Johnston, Jun 28 2011

digitSet[n_] := Union[IntegerDigits[n]]; Select[Range[0, 99000], digitSet[#] == digitSet[#^2] &] (* Jayanta Basu, Jun 02 2013 *)

isA029793(n)=Set(Vec(Str(n)))==Set(Vec(Str(n^2))) \\ Charles R Greathouse IV, Jun 28 2011

(0L to 99999L).filter(n => n.toString.toCharArray.toSet == (n * n).toString.toCharArray.toSet) // Alonso del Arte, Jan 19 2020

A095050 Numbers such that all ten digits are needed to write all positive divisors in decimal representation.

Original entry on oeis.org

108, 216, 270, 304, 306, 312, 324, 360, 380, 406, 432, 450, 504, 540, 570, 608, 612, 624, 630, 648, 654, 702, 708, 714, 720, 728, 756, 760, 780, 810, 812, 864, 870, 900, 910, 912, 918, 924, 936, 945, 954, 972, 980, 1008, 1014, 1026, 1032, 1036, 1038

Offset: 1

Reinhard Zumkeller, May 28 2004

A095048(a(n)) = 10.
Numbers n such that A037278(n), A176558(n) and A243360(n) contain 10 distinct digits. - Jaroslav Krizek, Jun 19 2014
Once a number is in the sequence, then all its multiples will be there too. The list of primitive terms begin: 108, 270, 304, 306, 312, 360, 380, ... - Michel Marcus, Jun 20 2014
Pandigital numbers A050278 and A171102 are subsequences. - Michel Marcus, May 01 2020

			Divisors of 108 are: [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108] where all digits can be found.

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

Cf. A095048, A059436 (subsequence), A206159.
Cf. A243543 (the smallest number m whose list of divisors contains n distinct digits).
Sequences of numbers n such that the list of divisors of n contains k distinct digits for 1 <= k <= 10: k = 1: A243534; k = 2: A243535; k = 3: A243536; k = 4: A243537; k = 5: A243538; k = 6: A243539; k = 7: A243540; k = 8: A243541; k = 9: A243542; k = 10: A095050. - Jaroslav Krizek, Jun 19 2014
Cf. A050278, A171102.

import Data.List (elemIndices)
a095050 n = a095050_list !! (n-1)
a095050_list = map (+ 1) $ elemIndices 10 $ map a095048 [1..]
-- Reinhard Zumkeller, Feb 05 2012

q:= n-> is({$0..9}=map(x-> convert(x, base, 10)[], numtheory[divisors](n))):
select(q, [$1..2000])[];  # Alois P. Heinz, Oct 28 2021

Select[Range@2000, 1+Union@@IntegerDigits@Divisors@# == Range@10 &] (* Hans Rudolf Widmer, Oct 28 2021 *)

isok(m)=my(d=divisors(m), v=[1]); for (k=2, #d, v = Set(concat(v, digits(d[k]))); if (#v == 10, return (1));); #v == 10; \\ Michel Marcus, May 01 2020

from sympy import divisors
def ok(n):
    digits_used = set()
    for d in divisors(n):
        digits_used |= set(str(d))
    return len(digits_used) == 10
print([k for k in range(1040) if ok(k)]) # Michael S. Branicky, Oct 28 2021

a(n) ~ n. - Charles R Greathouse IV, Nov 16 2022

A303948 A fractal-like sequence: erasing all pairs of consecutive terms that have at least one digit in common leaves the sequence unchanged.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 12, 30, 13, 22, 21, 33, 23, 10, 24, 14, 25, 15, 26, 16, 27, 17, 28, 18, 29, 19, 32, 31, 40, 34, 11, 20, 35, 36, 12, 30, 41, 42, 13, 22, 37, 38, 21, 33, 44, 43, 50, 45, 23, 10, 24, 39, 49, 51, 52, 14, 25, 46, 47, 15, 26, 48, 54, 16, 27, 53, 55, 17, 28

Offset: 1

Eric Angelini and Lars Blomberg, May 03 2018

The sequence is fractal-like as it embeds an infinite number of copies of itself.
The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
  1) no overlapping pairs of parentheses;
  2) always start the content inside a pair of parentheses with the smallest integer S > 9 not yet present inside another pair of parentheses;
  3) always end the content inside a pair of parentheses with the smallest integer H > 9 not yet present inside another pair of parentheses such that the integers S and H have at least one digit in common;
  4) after a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 6, a(7) = 7, a(8) = 8, a(9) = 9, a(10) = 10, always try to extend the sequence with a duplicate > 9 of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses;
  5) Never use a term of A171102 (Pandigital numbers: numbers containing the digits 0-9. Version 2: each digit appears at least once).

			Parentheses are added around each pair of terms having at least one digit in common:
1,2,3,4,5,6,7,8,9,(10,11),(20,12),(30,13),(22,21),(33,23),10,(24,14),(25,15),(26,16),(27,17),(28,18),(29,19),(32,31),(40,34),11,20,(35,36),12,30,(41,42),13,
Erasing all the parenthesized contents yields
1,2,3,4,5,6,7,8,9,(.....),(.....),(.....),(.....),(.....),10,(.....),(.....),(.....),(.....),(.....),(.....),(.....),(.....),11,20,(.....),12,30,(.....),13,
We see that the remaining terms slowly rebuild the starting sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A303845 for another "erasing criterion" (prime by concatenation).

A051018 Numbers that are 2-persistent but not 3-persistent.

Original entry on oeis.org

1023456789, 1023456879, 1023457689, 1023457869, 1023458679, 1023458769, 1023465789, 1023465879, 1023467589, 1023467859, 1023468579, 1023468759, 1023475689, 1023475869, 1023476589, 1023476859, 1023478569, 1023478659, 1023485679, 1023485769, 1023486579, 1023486759

Offset: 1

Eric W. Weisstein

A number m is k-persistent iff all of {m, 2m,..., km} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (pandigital), A204047 (smallest n-persistent), A051264 (1-persistent), A051019 (3-persistent), A051020 (4-persistent), A204096 (5-persistent), A204097 (6-persistent).

is_A051018(n,k=3)=10>#Set(Vec(Str(k*n))) & !while(k--,9<#Set(Vec(Str(k*n))) || return(0)) \\ M. F. Hasler, Jan 10 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

A051019 Numbers that are 3-persistent but not 4-persistent.

Original entry on oeis.org

1052674893, 1052687493, 1052746893, 1052748693, 1052867493, 1052874693, 1053267489, 1053268749, 1053274869, 1053286749, 1053287469, 1065273489, 1065287349, 1067285493, 1067328549, 1068547293, 1068547329, 1068549273

Offset: 1

Eric W. Weisstein

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (pandigital), A204047 (smallest n-persistent), A051264 (1-persistent), A051018 (2-persistent), A051020 (4-persistent), A204096 (5-persistent), A204097 (6-persistent).

is_A051019(n)=for(i=1,4, 9<#Set(Vec(Str(i*n))) || return(i>3)) \\ M. F. Hasler, Jan 10 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

A051020 Numbers that are 4-persistent but not 5-persistent.

Original entry on oeis.org

1053274689, 1089467253, 1253094867, 1267085493, 1268547309, 1269085473, 1273085469, 1308547269, 1308549267, 1326854907, 1327068549, 1328746905, 1450687329, 1450732869, 1450867293, 1450928673, 1452687309, 1452690873

Offset: 1

Eric W. Weisstein

A number n is k-persistent iff all of {n, 2*n, ..., k*n} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, Apr 28 2022)
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (pandigital), A204047 (smallest n-persistent), A051264 (1-persistent), A051018 (2-persistent), A051019 (3-persistent), A204096 (5-persistent), A204097 (6-persistent).

is_A051020(n)=for(i=1, 5, 9<#Set(Vec(Str(i*n))) || return(i>4)) \\ M. F. Hasler, Jan 10 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

Sequence corrected by Hans Havermann, Jan 11 2012

A051264 Numbers that are 1-persistent but not 2-persistent.

Original entry on oeis.org

1023456798, 1023456897, 1023456978, 1023456987, 1023457698, 1023457896, 1023457968, 1023457986, 1023458697, 1023458796, 1023458967, 1023458976, 1023459678, 1023459687, 1023459768, 1023459786, 1023459867, 1023459876

Offset: 1

Eric W. Weisstein

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Hans Havermann, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Persistent Number

Cf. A171102 (1-persistent, i.e. pandigital numbers), A204047 (smallest n-persistent), A051018 (2-persistent), A051019 (3-persistent), A051020 (4-persistent), A204096 (5-persistent), A204097 (6-persistent).

A051264 = A171102 \ A051018. - M. F. Hasler, Jan 09 2012

Definition corrected by Franklin T. Adams-Watters, Jan 09 2012

A134336 Nonnegative integers n containing each digit between n's smallest and largest decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 201, 210, 211, 212, 213, 221, 222, 223, 231, 232, 233, 234, 243, 312, 321, 322, 323

Offset: 1

Rick L. Shepherd, Oct 21 2007

A032981 is a subsequence; the term 102 is the first positive integer not also in A032981. A171102 (pandigital numbers) and A033075 are subsequences. Union of A010785 (repdigits) and A108965.
a(n) = A178403(n) for n < 48. - Reinhard Zumkeller, May 27 2010
Equivalently, numbers with the property that the set of its decimal digits is a set of consecutive numbers. - Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A032981, A050278, A033075 (a subsequence), A010785, A108965.

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8));for(i=2,#v,if(v[i]!=1+v[i-1],return(0)));1 \\ Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

is_A134336(n)={vecmax(n=Set(digits(n)))-vecmin(n)==#n-1} \\ M. F. Hasler, Dec 24 2014

def ok(n): d = sorted(set(map(int, str(n)))); return d[-1]-d[0]+1 == len(d)
print([k for k in range(324) if ok(k)]) # Michael S. Branicky, Dec 12 2023

a(n) ~ n. - Charles R Greathouse IV, Sep 09 2011

Edited by N. J. A. Sloane, Aug 06 2012

A204096 Numbers that are 5-persistent but not 6-persistent.

Original entry on oeis.org

13047685942, 14057869523, 20476859413, 21304768594, 35078196492, 35079219648, 35079648192, 35079649218, 35081964792, 35092196478, 35096478192, 35096479218, 35180796492, 35180964792, 35192079648, 35192096478, 35196478092, 35196479208

Offset: 1

Hans Havermann, Jan 10 2012

A number n is k-persistent iff all of {n, 2n,..., kn} are pandigital (in the sense of A171102).

Ross Honsberger, More Mathematical Morsels, Mathematical Association of America, 1991, pages 15-18.

Hans Havermann, Table of n, a(n) for n = 1..1000

Cf. A171102 (pandigital), A051264 (1-persistent), A051018 (2-persistent), A051019 (3-persistent), A051020 (4-persistent), A204097 (6-persistent), A204047 (smallest n-persistent).

cp's OEIS Frontend

A047726 Number of different numbers that are formed by permuting digits of n.

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A029793 Numbers k such that k and k^2 have the same set of digits.

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A095050 Numbers such that all ten digits are needed to write all positive divisors in decimal representation.

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A303948 A fractal-like sequence: erasing all pairs of consecutive terms that have at least one digit in common leaves the sequence unchanged.

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A051018 Numbers that are 2-persistent but not 3-persistent.

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A051019 Numbers that are 3-persistent but not 4-persistent.

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A051020 Numbers that are 4-persistent but not 5-persistent.

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