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

1,2

Comments

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.

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)

Example

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

Crossrefs

Sequence in context: A283870 A175688 A048321 * A295010 A257054 A110784

Adjacent sequences: A071264 A071265 A071266 * A071268 A071269 A071270

Keyword

base,nonn

Author

Amarnath Murthy, Jun 01 2002

Extensions

Corrected and extended by Diana L. Mecum, Jul 06 2007

Status

approved