cp's OEIS Frontend

A379736 Number of integer partitions of n whose product of parts is not n.

%I A379736 #5 Jan 08 2025 09:28:28
%S A379736 1,0,1,2,3,6,9,14,19,28,40,55,73,100,133,174,226,296,381,489,623,790,
%T A379736 1000,1254,1568,1956,2434,3007,3714,4564,5599,6841,8342,10141,12308,
%U A379736 14881,17968,21636,26013,31183,37331,44582,53169,63260,75171,89130,105556
%N A379736 Number of integer partitions of n whose product of parts is not n.
%C A379736 These partitions are ranked by A379722, complement A301987.
%F A379736 a(n) = A000041(n) - A001055(n).
%e A379736 The a(2) = 1 through a(7) = 14 partitions:
%e A379736   (11)  (21)   (31)    (32)     (33)      (43)
%e A379736         (111)  (211)   (41)     (42)      (52)
%e A379736                (1111)  (221)    (51)      (61)
%e A379736                        (311)    (222)     (322)
%e A379736                        (2111)   (411)     (331)
%e A379736                        (11111)  (2211)    (421)
%e A379736                                 (3111)    (511)
%e A379736                                 (21111)   (2221)
%e A379736                                 (111111)  (3211)
%e A379736                                           (4111)
%e A379736                                           (22111)
%e A379736                                           (31111)
%e A379736                                           (211111)
%e A379736                                           (1111111)
%t A379736 Table[Length[Select[IntegerPartitions[n],Times@@#!=n&]],{n,0,30}]
%Y A379736 The complement is counted by A001055.
%Y A379736 The strict case is A111133 (except first term).
%Y A379736 A000041 counts integer partitions, strict A000009.
%Y A379736 A002865 counts partitions into parts > 1, see A379734, strict A379735.
%Y A379736 A324851 finds numbers > 1 divisible by the sum of their prime indices.
%Y A379736 A379666 counts partitions by sum and product, without 1's A379668.
%Y A379736 Counting and ranking multisets by comparing sum and product:
%Y A379736 - same: A001055, ranks A301987
%Y A379736 - divisible: A057567, ranks A326155
%Y A379736 - divisor: A057568, ranks A326149, see A379733
%Y A379736 - greater than: A096276 shifted right, ranks A325038
%Y A379736 - greater or equal: A096276, ranks A325044
%Y A379736 - less than: A114324, ranks A325037, see A318029, A379720
%Y A379736 - less or equal: A319005, ranks A379721, see A025147
%Y A379736 - different: A379736 (this), ranks A379722
%Y A379736 Cf. A028422, A069016, A319000, A319916, A325036, A325041, A326152, A379671, A379678.
%K A379736 nonn
%O A379736 0,4
%A A379736 _Gus Wiseman_, Jan 07 2025