A045876 -id:A045876 - cp's OEIS frontend

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

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

A275945 Numbers n such that the average of different permutations of digits of n is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 1

Altug Alkan, Aug 29 2016

Complement of A273492.
Permutations with a first digit of 0 are included in the average (i.e. 0010 is taken to be 10, 01 is taken to be 1, etc.).
From Robert Israel, Sep 01 2016: (Start)
n such that A002275(A055642(n))*A007953(n) is divisible by A055642(n).
In particular, contains all k-digit numbers if k is in A014950. (End)

			97 is a term because (97+79) is divisible by 2.
100 is a term because (1+10+100) is divisible by 3.
123 is a term because (123+132+213+231+312+321) is divisible by 6.
1001 is not a term because (11+101+110+1001+1010+1100) is not divisible by 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002275, A014950, A045876, A055642, A066642, A007953, A273492.

f:= proc(n) local L, d, s;
  L:= convert(n, base, 10);
  d:= nops(L);
  s:= convert(L, `+`);
  evalb(s*(10^d-1)/9 mod d = 0)
end proc:
select(f, [$1..10000]); # Robert Israel, Sep 01 2016

Select[Range@ 111, IntegerQ@ Mean@ Map[FromDigits, Permutations@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Aug 29 2016 *)

A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
for(n=1, 2000, if((((10^A055642(n)-1)/9)*A007953(n)) % A055642(n) == 0, print1(n, ", ")));

A273492 Numbers n such that the average of different permutations of digits of n is not an integer.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 1000, 1001, 1002, 1004, 1005, 1006, 1008, 1009, 1010, 1011, 1013, 1014, 1015, 1017, 1018, 1019, 1020, 1022, 1023, 1024

Offset: 1

Altug Alkan, Aug 29 2016

Complement of A275945.
Permutations with a first digit of 0 are included in the average (i.e. 0010 is taken to be 10, 01 is taken to be 1, etc.).
From Robert Israel, Sep 01 2016: (Start)
n such that A002275(A055642(n))*A007953(n) is not divisible by A055642(n).
In particular, contains no k-digit numbers if k is in A014950. (End)

			12 is a term because (12+21) = 33 is not divisible by 2.
1000 is a term because (1+10+100+1000) = 1111 is not divisible by 4.
123 is not a term because (123+132+213+231+312+321) is divisible by 6.
1001 is a term because (11+101+110+1001+1010+1100) is not divisible by 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002275, A014950, A055642, A066642, A007953, A045876, A055642, A275945.

f:= proc(n) local L,d,s;
  L:= convert(n,base,10);
  d:= nops(L);
  s:= convert(L,`+`);
  evalb(s*(10^d-1)/9 mod d = 0)
end proc:
remove(f, [$1..10000]); # Robert Israel, Sep 01 2016

Select[Range[2^10], ! IntegerQ@ Mean@ Map[FromDigits, Permutations@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Aug 29 2016 *)

A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
for(n=1, 2000, if((((10^A055642(n)-1)/9)*A007953(n)) % A055642(n) != 0, print1(n, ", ")));

A276354 Palindromes n > 0 such that the sum of all distinct permutations of the digits of n is also a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 191, 202, 212, 222, 232, 242, 252, 272, 303, 313, 323, 333, 353, 404, 414, 434, 444, 515, 555, 666, 777, 787, 868, 888, 949, 999, 1001, 1111, 1221, 2002, 2112, 2222, 2992

Offset: 1

Altug Alkan, Sep 05 2016

Values of A002113(n) such that A045876(A002113(n)) is in A002113 (n > 0).
If n has a zero digit then the permutations starting with 0 are included in the sum (i.e., 0010 is taken to be 10, 01 is taken to be 1, etc.).
A010785(n) is an obvious term of this sequence for all n > 0.

			101 is a term because 11 + 101 + 110 = 222 is also a palindrome.
232 is a term because 223 + 232 + 322 = 777 is also a palindrome.
2002 is a term because 22 + 202 + 220 + 2002 + 2020 + 2200 = 6666 is also a palindrome.

Indranil Ghosh, Table of n, a(n) for n = 1..385

Cf. A002113, A007953, A010785, A045876, A047726, A055642.

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA002113(n) = n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1;
lista(nn) = for(n=1, nn, if(isA002113(n) && isA002113(A045876(n)), print1(n", ")));

A276590 Pandigital numbers n such that sum of all permutations of digits of n is also a pandigital number. Sequence lists the least ones of corresponding permutational classes.

Original entry on oeis.org

10234567889, 100223456789, 100234566789, 100234567889, 101234556789, 101234567789, 102234456789, 102234566789, 102334556789, 102334567899, 102344456789, 102344567889, 102345567789, 102345666789, 102345677789, 102345678899, 1000223456789, 1000234456789, 1000234566789

Offset: 1

Robert Israel and Altug Alkan, Sep 06 2016

The least pandigital number in A276510 is 10234567889.
Each member of the sequence has digits in increasing order except that the first digit is 1.
The sequence has 1 member with 11 digits, 15 with 12 digits, 90 with 13 digits, 261 with 14 digits and 1190 with 15 digits.

			100223456789 is a term because A045876(100223456789) = 52113599999947886400 is pandigital.

Robert Israel, Table of n, a(n) for n = 1..1557

Cf. A045876, A171102, A276510.

pandig:= n -> evalb(nops(convert(convert(n,base,10),set))=10):
sump:= 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:
n0:= 1023456789:
rep:= proc(n) local L,n0,i;
  L:= sort(convert(n,base, 10));
  n0:= numboccur(0, L);
  L:= subsop(1=1,n0+1=0,L);
  add(L[-i-1]*10^(i),i=0..nops(L)-1); end proc:
sort(convert(map(rep,
select(pandig @ sump, {seq(seq(n0*10^d+x,x=0..10^d-1),d=0..3)})),list));

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

Original entry on oeis.org

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);

A276596 Least k such that A276502(k) = n.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 101, 1000000, 1001

Offset: 1

Altug Alkan, Sep 11 2016

Cf. A276502.

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
A276502(n) = {my(k = 1); while (A045876(n*(10^k)) % A045876(n), k++); k; }
a(n) = {my(k = 1); while (A276502(k) != n, k++); k; }

cp's OEIS Frontend

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

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A275945 Numbers n such that the average of different permutations of digits of n is an integer.

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A273492 Numbers n such that the average of different permutations of digits of n is not an integer.

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A276354 Palindromes n > 0 such that the sum of all distinct permutations of the digits of n is also a palindrome.

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A276590 Pandigital numbers n such that sum of all permutations of digits of n is also a pandigital number. Sequence lists the least ones of corresponding permutational classes.

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

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A276596 Least k such that A276502(k) = n.

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