A119646 a(n) = maximum number of partitions of n into 3 parts, each having the same product.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3

Offset

3,11

Links

Table of n, a(n) for n=3..108.

Example

a(3)=1, because there is only one way to partition 3.
a(13)=2, because 13 = 6+6+1 = 9+2+2 and 6*6*1 = 9*2*2 = 36.
a(39)=3, because 39 = 20+15+4 = 24+10+5 = 25+8+6 and 20*15*4 = 24*10*5 = 25*8*6 = 1200.
See A103277 for more examples.

Mathematica

pdt[lst_] := lst[[1]]*lst[[2]]*lst[[3]];
tanya[n_] := Max[Length /@ Split[Sort[pdt /@ Union[ Partition[Last /@ Flatten[ FindInstance[a + b + c == n && a >= b >= c > 0, {a, b, c}, Integers, (* failsafe *) Round[n^2/12]]], 3]] ]]];
Table[ tanya@n, {n, 4, 108}]
Table[SortBy[Tally[Times@@@IntegerPartitions[n, {3}]], Last][[-1, 2]], {n, 3, 110}] (* Harvey P. Dale, Jan 08 2023 *)

Crossrefs

Cf. A103277, A119028.

Sequence in context: A230866 A158209 A234538 * A024693 A025126 A129706

Adjacent sequences: A119643 A119644 A119645 * A119647 A119648 A119649

Keyword

nonn

Author

Joseph Biberstine (jrbibers(AT)indiana.edu) and Robert G. Wilson v, Jul 27 2006

Extensions

Name clarified by Dmitry Kamenetsky, Aug 02 2015

Status

approved