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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. 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 A364274 A118764
KEYWORD
easy,nonn,base,look
AUTHOR
STATUS
approved

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Last modified May 18 15:24 EDT 2024. Contains 372664 sequences. (Running on oeis4.)