A140437 - OEIS

%I #9 Aug 01 2015 13:50:39

%S 1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,9,9,9,10,

%T 11,12,13,14,16,18,19,21,24,26,28,30,31,36,38,41,44,49,51,54,60,65,70,

%U 76,81,89,93,102,111,120,131,144,155,167,182,201,216,236,254,279,303,336,363,402,431,476

%N a(n) is the maximal number of partitions of n of the same length with the same product.

%C This sequence was inspired by John Conway's Wizards puzzle (see link).

%H Tanya Khovanova, <a href="http://blog.tanyakhovanova.com/?p=29">John Conway's Wizards Puzzle</a>

%e The number 13 can be partitioned into 3 numbers with the same product in two ways: {1,6,6} and {2,2,9}. It can also be partitioned into 5 numbers with the same product in two ways: {1,1,3,4,4} and {1,2,2,2,6}. 13 can't have 3 different partitions of the same length with the same product. Hence a(13) = 2.

%t Table[Max[ Transpose[ Flatten[Table[ Tally[Apply[Times, IntegerPartitions[k, {n}], 2]], {n, k}], 1]][[2]]], {k, 60}]

%t Table[ Max[ Transpose[ Flatten[ Table[ Tally[ Apply[ Times, IntegerPartitions[k, {n}], 2]], {n, k}], 1]][[2]]], {k, 60}] (* _Robert G. Wilson v_, Aug 19 2008 *)

%K nonn

%O 1,12

%A _Tanya Khovanova_, Jun 20 2008, Jun 23 2008

%E More terms from _Robert G. Wilson v_, Aug 19 2008