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A045876 Sum of different permutations of digits of n (leading 0's allowed). 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 a(n) = A*N*I = (19/6)*(360)*(111111) = 126666540. - Amarnath Murthy, Jul 14 2003

It seems that the first person who has 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)

FORMULA

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

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 *)

PROG

(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

CROSSREFS

Same beginning as A033865. Cf. A061147.

Sequence in context: A265558 A082273 A256755 * A033865 A118764 A226134

Adjacent sequences:  A045873 A045874 A045875 * A045877 A045878 A045879

KEYWORD

easy,nonn,base,look

AUTHOR

Erich Friedman

STATUS

approved

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Last modified November 13 02:59 EST 2019. Contains 329085 sequences. (Running on oeis4.)