A071267 - OEIS

%I #16 Jan 28 2023 12:18:24

%S 1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,110,111,121,132,143,154,

%T 165,176,187,222,333,444,555,666,777,888,999,1110,1111,1221,1332,1443,

%U 1554,1665,1776,1887,1998,2109,2220,2222,2331,2442,2553,2664,2775,2886

%N Numbers which can be expressed as the sum of all distinct digit permutations of some number k.

%C 222 can be expressed so in two different ways, i.e., 222 = 200 + 020 + 002 as well as 222 = 101 + 110 + 011. Problem: find a number which can be so expressed in n different ways.

%H David W. Wilson, <a href="/A071267/b071267.txt">Table of n, a(n) for n = 1..9450</a>

%F From _David W. Wilson_, Jul 12 2007: (Start)

%F Let f(n) be the sum of all permuted versions of n. Let

%F   s(n) = sum of digits of n.

%F   d(n) = number of digits of n.

%F   c_n(k) = number of occurrences of digit k in n.

%F   p(n) = Product_{k=0..9} c_n(k)!.

%F   r(n) = n-digit rep-1 number = (10^n-1)/n.

%F   t(n) = s(n)*(d(n)-1)!/p(n).

%F Then f(n) = t(n)*r(d(n)).

%F For example, if n = 314159, we get

%F   s(n) = 23

%F   d(n) = 6

%F   c_n = (0, 2, 0, 1, 1, 1, 0, 0, 0, 1)

%F   p(n) = Product_{k=0..9} c_n(k)! = 2

%F   r(d(n)) = r(6) = 111111

%F   t(n) = 23*120/2 = 1380

%F and

%F   f(314159) = 1380*11111 = 153333180. (End)

%e 1110 is a term as it is the sum of all distinct permutations of 104, i.e., 104+140+410+401+014+041 = 1110.

%K base,nonn

%O 1,2

%A _Amarnath Murthy_, Jun 01 2002

%E Corrected and extended by _Diana L. Mecum_, Jul 06 2007