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 No big digits involved (Posted on 2017-11-06)
How many decimal integers below 10^20 exist having sum of their digits less than 4?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 conjecture | Comment 1 of 2
There are 3 1-digit numbers  1,2,3

6 2-digit numbers 10,11,12,20,21,30

10 3-digit numbers  100,101,102,110,111,120,200,201,210,300

15 4-digit numbers 1000,1001,1002,1010,1011,1020,1100,1101,1200,2000,2001,2010,2100

Based on this evidence I conjecture there are (n+1)*(n+2)/2 n-digit numbers of the desired type.  If that's the case the answer is the sum of 3 + 6 + 10 + 15 +. . .+ 231.

Checking the differences I get the formula (n/6)*(n^2 + 6n + 11) with value = 1770 for n = 20.  A quick run of the tape confirms.

 Posted by xdog on 2017-11-06 17:33:56

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