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

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

Author

Erich Friedman

Keywords

• easy
• nonn
• base
• look
• changed

Comments

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

References

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

Links

• 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)

Crossrefs

Same beginning as A033865. Cf. A061147.

Programs

Maple

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

Mathematica

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

PARI

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

PARI

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

Formula

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