Talk:Minimum multiple of m where digital sum equals m: Difference between revisions
Talk:Minimum multiple of m where digital sum equals m (view source)
Revision as of 17:20, 19 February 2022
, 2 years ago→I'm out
(some preliminary brute force attack results) |
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The first potential 14 digit candidate is 11999999999999 (119999999999990*, 15 digits)
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==165==
I thought it might be useful to work 165. The minimal starting value is 7999999999999999995 as expressed below:
21111111111
09876543210987654321
--------------------
7999999999999999995
let Ʃo = sum of odd digits = n<sub>19</sub>+n<sub>1</sub>+No where No = sum of n<sub>3</sub> .. 2 .. n<sub>17</sub> <br>
let Ʃe = sum of even digits = n<sub>20</sub>+Ne where Ne = sum of n<sub>2</sub> .. 2 .. n<sub>18</sub>
n<sub>1</sub> = 5
Now consider the table:
n<sub>20</sub> n<sub>19</sub> No Ne Ʃo Ʃe (Ʃo-Ʃe % 11 = 0)
0 7 72 81 84 81 No
0 8 71 81 84 81 No
0 8 72 80 85 80 No
0 9 70 81 84 81 No
0 9 71 80 85 80 No
0 9 71 80 85 80 No
0 9 72 79 86 79 No
1 6 72 81 83 82 No
1 7 71 81 83 82 No
1 7 72 80 84 81 No
1 8 70 81 83 82 No
1 8 71 80 84 81 No
1 8 71 80 84 81 No
1 8 72 79 85 80 No
1 9 69 81 83 82 No
1 9 70 80 84 81 No
1 9 70 80 84 81 No
1 9 71 79 85 80 No
1 9 70 80 84 81 No
1 9 71 79 85 80 No
1 9 71 79 85 80 No
1 9 72 78 86 79 No
2 5 72 81 82 83 No
.
.
.
7 0 72 81 77 88 Yes
n<sub>20</sub>+n<sub>19</sub> must be at least 7. Each time n<sub>20</sub>+n<sub>19</sub> is incremented 1 must be subtracted from either No or Ne. The table above depicts the possibilities and is in fact a perfectly balanced binary tree. When a candidate is identified No and Ne must be expanded to the set of 8 digits summing to No and the set of 9 digits summing to Ne, and all combinations considered. In this case it is easy as 72 can only be 8 nines and 81 can only be 9 nines. Thus it only remains to prove that 70999999999999999995 is divisible by 3.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 13:37, 8 February 2022 (UTC)
==I'm out==
Gave up on 370. 275 still a little tardy/not done 25. Pretty pleased with 200 in 0.7s though. --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 12:21, 14 February 2022 (UTC)
:I'm trying to decide on an adjective to describe this. I might have to toss a coin to choose between insane and impressive.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 13:41, 14 February 2022 (UTC)
::I'm insanely impressed. Chapeau! Kudos to Pete :-) --[[User:Horst.h|Horst.h]] ([[User talk:Horst.h|talk]]) 16:14, 14 February 2022 (UTC)
::: FWIW, you may like to consider submitting this to OEIS for inclusion on the [[oeis:A131382|A131382]] page. They only have a list of the first 90 terms at the moment. If you are so inclined. --[[User:Thundergnat|Thundergnat]] ([[User talk:Thundergnat|talk]]) 16:56, 14 February 2022 (UTC)
::: Good idea, done. Update: that page links to [[oeis:A002998|A002998]] which aleady had '''1000''' terms (gulp) and a c++ program... --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 15:59, 15 February 2022 (UTC)
::: OK, now I really, really, really am out ''(please, pretty please!)'' --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 17:19, 19 February 2022 (UTC)
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