cp's OEIS Frontend

A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

%I A379735 #8 Jan 07 2025 21:20:53
%S A379735 0,1,1,1,1,1,1,1,2,2,1,3,1,3,4,4,1,8,1,11,9,7,1,26,7,10,18,33,1,67,1,
%T A379735 56,37,20,69,158,1,27,70,252,1,280,1,207,402,52,1,834,133,423,226,465,
%U A379735 1,1132,635,1541,388,129,1,3377,1,171,2891,3561,1674,3154
%N A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.
%C A379735 These partitions are ranked by the odd squarefree terms of A326149.
%e A379735 The a(n) partitions for n = 2, 9, 12, 15, 18, 20, 21:
%e A379735   (2)  (9)    (12)     (15)      (18)       (20)         (21)
%e A379735        (6,3)  (5,4,3)  (6,5,4)   (12,6)     (8,7,5)      (8,7,6)
%e A379735               (6,4,2)  (7,5,3)   (9,5,4)    (10,6,4)     (9,7,5)
%e A379735                        (10,3,2)  (9,6,3)    (10,8,2)     (11,7,3)
%e A379735                                  (9,7,2)    (11,5,4)     (12,7,2)
%e A379735                                  (6,5,4,3)  (12,5,3)     (14,4,3)
%e A379735                                  (7,6,3,2)  (7,6,5,2)    (7,6,5,3)
%e A379735                                  (9,4,3,2)  (8,5,4,3)    (9,7,3,2)
%e A379735                                             (9,5,4,2)    (7,5,4,3,2)
%e A379735                                             (10,5,3,2)
%e A379735                                             (6,5,4,3,2)
%t A379735 Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]
%Y A379735 Allowing 1's gives A379733.
%Y A379735 The non-strict version is A379734, allowing 1's A057568.
%Y A379735 A000041 counts integer partitions, strict A000009.
%Y A379735 A002865 counts partitions into parts > 1.
%Y A379735 A379666 counts partitions by sum and product, without 1's A379668.
%Y A379735 Counting and ranking multisets by comparing sum and product:
%Y A379735 - same: A001055, ranks A301987
%Y A379735 - divisible: A057567, ranks A326155
%Y A379735 - divisor: A057568, ranks A326149, see A379733
%Y A379735 - greater than: A096276 shifted right, ranks A325038
%Y A379735 - greater or equal: A096276, ranks A325044
%Y A379735 - less than: A114324, ranks A325037, see A318029, A379720
%Y A379735 - less or equal: A319005, ranks A379721, see A025147
%Y A379735 - different: A379736, ranks A379722, see A111133
%Y A379735 Cf. A069016, A319000, A319057, A319916, A324851, A326152, A379671, A379678.
%K A379735 nonn
%O A379735 1,9
%A A379735 _Gus Wiseman_, Jan 07 2025