A047726 -id:A047726 - cp's OEIS frontend

A276716 a(n) = A007953(n)*A047726(n).

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 6, 8, 10, 12, 14, 16, 18, 20, 4, 6, 4, 10, 12, 14, 16, 18, 20, 22, 6, 8, 10, 6, 14, 16, 18, 20, 22, 24, 8, 10, 12, 14, 8, 18, 20, 22, 24, 26, 10, 12, 14, 16, 18, 10, 22, 24, 26, 28, 12, 14, 16, 18, 20, 22, 12, 26, 28, 30, 14, 16, 18, 20, 22, 24, 26, 14, 30, 32, 16

Offset: 1

Altug Alkan, Sep 17 2016

Altug Alkan, Table of n, a(n) for n = 1..100000

Cf. A007953, A045876, A047726.

A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A007953(n) = sumdigits(n);
a(n) = A007953(n)*A047726(n);

A196222 a(n) = A047726(n)^A047726(n), where A047726(n) is the number of numbers which can be written with n's digits.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 27, 27, 46656

Offset: 0

M. F. Hasler, Sep 29 2011

Suggested by Franklin T. Adams-Watters and N. J. A. Sloane in analogy to A138908.

Cf. A138902, A138908.

A043537 Number of distinct base-10 digits of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(A000079(A130694(n))) = 10. - Reinhard Zumkeller, Jul 29 2007
a(A000290(A016070(n))) = 2. - Reinhard Zumkeller, Aug 05 2010
a(n) = 10 for almost all n. - Charles R Greathouse IV, Oct 02 2013

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

Cf. A047726, A118668, A119797, A119798, A119799, A119823, A038378, A076493, A055642, A178788.

import Data.List (nub)
a043537 = length . nub . show  -- Reinhard Zumkeller, Apr 16 2011

A043537 := proc(n)
        convert(convert(n,base,10),set) ;
        nops(%) ;
end proc: # R. J. Mathar, Dec 22 2012

Count[DigitCount@ #, n_ /; n > 0] & /@ Range@ 120 (* Michael De Vlieger, Aug 29 2015 *)

a(n)=#vecsort(digits(n),,8) \\ Charles R Greathouse IV, Oct 02 2013

def A043537(n): return len(set(str(n))) # Dimiter Skordev, Oct 02 2021

A179239 Permutation classes of integers, each identified by its smallest member.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 101, 102, 103

Offset: 0

Aaron Dunigan AtLee, Jul 04 2010

Let the "permutation set" of a positive integer n be the set of all integers formed by permuting the digits of n. Two integers are "permutationally congruent" if they generate the same permutation set. A "permutation class" is a set of all permutationally congruent integers. This sequence lists each permutation class, identified by its smallest member.
These are also the positive integers in order, omitting any d-digit number n if a previously listed d-digit number is a permutation of the digits of n.
Range of A328447: smallest representative of the equivalence class of all numbers having the same digits up to permutation. Equivalently: Numbers with digits in nondecreasing order, except that the smallest nonzero digit must precede the zero digits. This sequence is useful when considering functions which depend only on the digits of n, e.g., the number of primes contained in n, cf. A039993, A039999, A075053 and the records therein, A072857 (primeval numbers) and A076497, resp. A239196 and A239197, etc. - M. F. Hasler, Oct 18 2019

			The permutation set of 24 is {24, 42}, and this is the equivalence class modulo permutations of both of them, so 24 is listed, but 42 is not.
The permutation set of 30 is {3, 30}, but 3 is not in the same permutation class as 30 since 30 cannot be obtained by permuting digits of 3. Therefore 30 is listed separately from 3.
The numbers 89 and 98 are also permutationally congruent and form a permutation class, so only the smaller one is listed.

Aaron Dunigan AtLee and Michael De Vlieger, Table of n, a(n) for n = 0..10000 (first 2997 terms from Aaron Dunigan AtLee; prefix 0 by _Georg Fischer_, Oct 24 2019)

A variant of A009994.
Cf. A047726, A035927 (Number of distinct n-digit numbers up to permutations of digits).
Cf. A004186, A328447: largest & smallest representative of the class of n.

maxTerm = 103; (*maxTerm is the greatest term you wish to see*) permutationSet[n_Integer] := FromDigits /@ Permutations[IntegerDigits[n]]; permutationCongruentQ[x_Integer, y_Integer] := Sort[permutationSet[x]] == Sort[permutationSet[y]]; DeleteDuplicates[Range[maxTerm], permutationCongruentQ]
f[n_] := Block[{a = {0}, b = {DigitCount[0]}, i, w}, Do[w = DigitCount@ i; AppendTo[b, w]; If[! MemberQ[Most@ b, w], AppendTo[a, i]], {i, n}]; Rest@ a]; f@ 103 (* or faster: *)
Select[Range@ 103, LessEqual @@ IntegerDigits@ # || And[Take[IntegerDigits@ #, Last@ DigitCount@ # + 1] == Reverse@ Take[Sort@ IntegerDigits@ #, Last@ DigitCount@ # + 1], LessEqual @@ DeleteCases[IntegerDigits@ #, d_ /; d == 0]] &] (* Michael De Vlieger, Jul 14 2015 *)

is(n) = {my(d=digits(n),i); for(i=2,#d, if(d[i]!=0, d=vecextract(d,concat([1],vector(#d-i+1,j,i-1+j))); break));d==vecsort(d)||n/10^valuation(n,10)<10}
\\given an element n, in base b, find the next element from the sequence.
nxt(n,{b=10}) = {my(d = digits(n)); i = #d; while(i>0&&d[i]==b-1,i--); if(i>1, if(d[i]>0, d[i]++, d[i]=d[1];);for(j=i+1,#d,d[j]=d[i]), if(i==1, d[i]++;for(j=2,#d,d[j]=0), return(10^(#d))));sum(j=1,#d,d[j]*10^(#d-j))} \\ David A. Corneth, Apr 23 2016

select( is_A179239(n)={n==A328447(n)}, [0..200]) \\ M. F. Hasler, Oct 18 2019

from itertools import count, chain, islice
from sympy.utilities.iterables import combinations_with_replacement
def A179239_gen(): # generator of terms
    return chain((0,),(int(a+''.join(b)) for l in count(1) for a in '123456789' for b in combinations_with_replacement('0'+''.join(str(d) for d in range(int(a),10)),l-1)))
A179239_list = list(islice(A179239_gen(),31)) # Chai Wah Wu, Sep 13 2022

Prefixed with a(0) = 0 by M. F. Hasler, Oct 18 2019

A045876 Sum of different permutations of digits of n (leading 0's allowed).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 22, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 33, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 44, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 55, 121, 132, 143, 154, 66, 77, 88, 99, 110, 121, 66, 143

Offset: 1

Erich Friedman

Let the arithmetic mean of the digits of a 'D' digit number n be 'A', let 'N' = number of distinct numbers that can be formed by permuting the digits of n, and let 'I' = concatenation of 1 'D' times = (10^D-1)/9. then a(n) = A*N*I. E.g., let n = 324541, then A = (3+2+4+5+4+1)/6 = 19/6, N = 6!/(2!) = 360, I = 111111, and a(n) = A*N*I = (19/6)*(360)*(111111) = 126666540. - Amarnath Murthy, Jul 14 2003

It seems that the first person who studied the sum of different permutations of digits of a given number was the French scientist Eugène Aristide Marre (1823-1918). See links. - Bernard Schott, Dec 06 2012

Amarnath Murthy, An interesting result in combinatorics, Mathematics & Informatics Quarterly, Vol. 3, 1999, Bulgaria.

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.
A. Marre, Trouver la somme de toutes les permutations différentes d'un nombre donné., Nouvelles Annales de Mathématiques, 1ère série, tome 5 (1846), p. 57-60.
Norbert Verdier, QDV4 : Marre, Marre et Marre, page=1 (French mathematical forum les-mathematiques.net)

Same beginning as A033865. Cf. A061147.

f:= proc(x) local L,D,n,M,s,j;
  L:= convert(x,base,10);
  D:= [seq(numboccur(j,L),j=0..9)];
  n:= nops(L);
  M:= n!/mul(d!,d=D);
  s:= add(j*D[j+1],j=0..9);
  (10^n-1)*M/9/n*s
end proc:
map(f, [$1..100]); # Robert Israel, Jul 07 2015

Table[Total[FromDigits /@ Permutations[IntegerDigits[n]]], {n, 100}] (* T. D. Noe, Dec 06 2012 *)

A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
a(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n)); \\ Altug Alkan, Aug 29 2016

A045876(n) = {my(d=digits(n), q=1, v, t=1); v = vecsort(d); for(i=1, #v-1, if(v[i]==v[i+1], t++, q*=binomial(i, t); t=1)); q*binomial(#v, t)*(10^#d-1)*vecsum(d)/9/#d} \\ David A. Corneth, Oct 06 2016

a(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n)). - Altug Alkan, Aug 29 2016

A055098 Number of distinct anagrams of digits of n without leading zeros.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 4

Offset: 1

Henry Bottomley, Apr 19 2000

			a(101)=2 since the digits of 101 can be ordered 101 or 110 (but not 011).

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

Cf. A046810, A047726.

import Data.List (permutations, nub)
a055098 n = length $ nub $ filter ((> '0') . head) $ permutations $ show n
-- Reinhard Zumkeller, Aug 14 2011

a[n_] := Length[ DeleteCases[ Permutations[ IntegerDigits[n]], {0 .., }]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Nov 30 2011 *)

a(n)={my(v=digits(n), f=vector(10), n=#v); for(i=1, #v, f[1+v[i]]++); (1 - f[1]/n) * n! / prod(i=1, #f, f[i]!)} \\ Andrew Howroyd, Jan 27 2020

from math import factorial, prod
def a(n):
    s = str(n); d, c = len(s), [s.count(str(i)) for i in range(10)]
    return (d-c[0])*factorial(d-1)//prod(map(factorial, c))
print([a(n) for n in range(1, 50)]) # Michael S. Branicky, Aug 24 2022

a(n) = O(n/(log n)^(9/2)). - Charles R Greathouse IV, Aug 24 2022

A071268 Sum of all digit permutations of the concatenation of first n numbers.

Original entry on oeis.org

1, 33, 1332, 66660, 3999960, 279999720, 22399997760, 2015999979840, 201599999798400, 927359999990726400, 1064447999999893555200, 2058376319999997941623680, 4439635199999999955603648000, 10585935359999999998941406464000, 27655756127999999999972344243872000

Offset: 1

Amarnath Murthy, Jun 01 2002

The permutations yield n! different numbers and if they are stacked vertically then the sum of each column is (n-1)! times the n-th triangular number = (n-1)!*n(n+1)/2. a(n) = [(n+1)!/2]*[{10^n -1}/9]. Note that this is only valid for 1 <= n <= 9.

The first person who studied the sum of different permutations of digits of a given number seems to be the French scientist Eugène Aristide Marre (1823-1918). See links. - Bernard Schott, Dec 07 2012

			For n=3, a(3) = 123 + 132 + 213 + 231 + 312 + 321 = 1332. - _Michael B. Porter_, Aug 28 2016

Alois P. Heinz, Table of n, a(n) for n = 1..208, Jan 04 2019
A. Marre, Trouver la somme de toutes les permutations différentes d'un nombre donné., Nouvelles Annales de Mathématiques, 1ère série, tome 5 (1846), p. 57-60.
Norbert Verdier and Raymond Cordier, QDV4 : Marre, Marre et Marre, page=1 (French mathematical forum les-mathematiques.net)

Cf. A045876, A047726, A007908.

a:= proc(n) local s, t, l;
      s:= cat("", seq(i, i=1..n)); t:= length(s);
      l:= (p-> seq(coeff(p, x, i), i=0..9))(add(x^parse(s[i]), i=1..t));
      (10^t-1)/9*combinat[multinomial](t, l)*add(i*l[i+1], i=1..9)/t
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 04 2019

Table[Total@ Map[FromDigits, Permutations@ Flatten@ Map[IntegerDigits, Range@ n]], {n, 10}] (* or *)
Table[Function[d, (((10^Length@ d - 1)/9)* Length@ Union@ Map[FromDigits, Permutations@ d] Total[d])/Length@ d]@ Flatten@ Map[IntegerDigits, Range@ n], {n, 11}] (* Michael De Vlieger, Aug 30 2016, latter after Harvey P. Dale at A047726 *)

A007908(n) = my(s=""); for(k=1, n, s=Str(s, k)); eval(s);
A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
a(n) = ((10^A055642(A007908(n))-1)/9)*(A047726(A007908(n))*A007953(A007908(n))/(A055642(A007908(n)))); \\ Altug Alkan, Aug 28 2016

from math import factorial
from operator import mul
from functools import reduce
def A071268(n):
    s = ''.join(str(i) for i in range(1,n+1))
    return sum(int(d) for d in s)*factorial(len(s)-1)*(10**len(s)-1)//(9*reduce(mul,(factorial(d) for d in (s.count(w) for w in set(s))))) # Chai Wah Wu, Jan 04 2019

a(n) = (n + 1)!*(10^n - 1)/18 for 1 <= n <= 9.

a(n) = ((10^A055642(A007908(n))-1)/9)*(A047726(A007908(n))*A007953(A007908(n))/(A055642(A007908(n)))). - Altug Alkan, Aug 28 2016

Edited by Robert G. Wilson v, Jun 03 2002

Corrected by Altug Alkan, Aug 28 2016

A138902 a(n) = d!, where d is the number of digits in n.

Original entry on oeis.org

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

Offset: 0

Odimar Fabeny, May 16 2008

James Spahlinger, Table of n, a(n) for n = 0..10000

Different from A047726. Cf. A138908.

a138902 = a000142 . a055642 -- James Spahlinger, Oct 09 2012

a[n_]:=IntegerLength[n]!;Array[a,103,0] (* James C. McMahon, Jun 24 2025 *)

a(n) = A000142(A055642(n)). - James Spahlinger, Oct 09 2012

Edited by N. J. A. Sloane, Sep 29 2011, at the suggestion of Franklin T. Adams-Watters

A193459 Total number of distinct divisors of all numbers that can be written by rearranging the digits of n in decimal representation.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8, 3, 5, 6, 6, 3, 8, 5, 6, 8, 4, 7, 12, 8, 6, 13, 8, 7, 8, 3, 7, 4, 5, 5, 12, 3, 5, 6, 8, 5, 12, 5, 6, 11, 9, 5, 16, 6, 6, 6, 8, 5, 11, 4, 11, 8, 7, 5, 12, 6, 6, 12, 9, 11, 8, 7, 8, 14, 8, 3, 13, 3, 5, 8, 7, 4, 10, 3, 10, 8

Offset: 1

Reinhard Zumkeller, Jul 26 2011

a(n) >= A000005(n), a(A193460(n)) = A000005(A193460(n)).

			a(20) = #{1,2,4,5,10,20} = 6;
a(21) = #{1,2,3,4,6,7,12,21} = 8;
a(22) = #{1,2,11,22} = 4;
a(23) = #{1,2,4,8,16,23,32} = 7;
a(24) = #{1,2,3,4,6,7,8,12,14,21,24,42} = 12;
a(25) = #{1,2,4,5,13,25,26,52} = 8;
a(26) = #{1,2,13,26,31,62} = 6;
a(27) = #{1,2,3,4,6,8,9,12,18,24,27,36,72} = 13;
a(28) = #{1,2,4,7,14,28,41,82} = 8;
a(29) = #{1,2,4,23,29,46,92} = 7.

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

Cf. A047726.

import Data.List (permutations, nub)
a193459 n =
   length $ nub $ concatMap divisors $ map read $ permutations $ show n
      where divisors n = filter ((== 0) . mod n) [1..n]
a193459_list = map a193459 [1..]

A277061 Numbers with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 115

Offset: 1

J. Lowell, Sep 26 2016

Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.

To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - David A. Corneth, Sep 27 2016

			25 is not in this sequence because 2*5 = 10 and 1*0 = 0.

Index entries for 10-automatic sequences.

Cf. A009994, A047726.
Cf. A031347, A034048 (complement).
Cf. A028843 (a subsequence).
Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
Cf. A052382 (a supersequence).

Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Sep 26 2016 *)

is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ David A. Corneth, Sep 27 2016

More terms from Michael De Vlieger, Sep 26 2016

cp's OEIS Frontend

A276716 a(n) = A007953(n)*A047726(n).

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A196222 a(n) = A047726(n)^A047726(n), where A047726(n) is the number of numbers which can be written with n's digits.

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A043537 Number of distinct base-10 digits of n.

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A179239 Permutation classes of integers, each identified by its smallest member.

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A045876 Sum of different permutations of digits of n (leading 0's allowed).

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A055098 Number of distinct anagrams of digits of n without leading zeros.

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A071268 Sum of all digit permutations of the concatenation of first n numbers.

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