A071267 Numbers which can be expressed as the sum of all distinct digit permutations of some number k.
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111, 121, 132, 143, 154, 165, 176, 187, 222, 333, 444, 555, 666, 777, 888, 999, 1110, 1111, 1221, 1332, 1443, 1554, 1665, 1776, 1887, 1998, 2109, 2220, 2222, 2331, 2442, 2553, 2664, 2775, 2886
Offset: 1
Examples
1110 is a term as it is the sum of all distinct permutations of 104, i.e., 104+140+410+401+014+041 = 1110.
Links
- David W. Wilson, Table of n, a(n) for n = 1..9450
Formula
From David W. Wilson, Jul 12 2007: (Start)
Let f(n) be the sum of all permuted versions of n. Let
s(n) = sum of digits of n.
d(n) = number of digits of n.
c_n(k) = number of occurrences of digit k in n.
p(n) = Product_{k=0..9} c_n(k)!.
r(n) = n-digit rep-1 number = (10^n-1)/n.
t(n) = s(n)*(d(n)-1)!/p(n).
Then f(n) = t(n)*r(d(n)).
For example, if n = 314159, we get
s(n) = 23
d(n) = 6
c_n = (0, 2, 0, 1, 1, 1, 0, 0, 0, 1)
p(n) = Product_{k=0..9} c_n(k)! = 2
r(d(n)) = r(6) = 111111
t(n) = 23*120/2 = 1380
and
f(314159) = 1380*11111 = 153333180. (End)
Extensions
Corrected and extended by Diana L. Mecum, Jul 06 2007
Comments