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A071267
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Numbers which can be expressed as the sum of all distinct digit permutations of some number k.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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)
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EXAMPLE
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1110 is a term as it is the sum of all distinct permutations of 104, i.e., 104+140+410+401+014+041 = 1110.
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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