cp's OEIS Frontend

A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

%I A380343 #7 Jan 23 2025 00:19:02
%S A380343 1,0,0,0,0,1,0,1,0,1,0,3,0,3,5,5,0,8,0,15,11,8,0,42,8,12,26,49,0,100,
%T A380343 0,90,56,27,105,246,0,41,108,414,0,450,0,332,651,81,0,1341,210,693,
%U A380343 366,754,0,1869,1044,2579,634,206,0,5695,0,278,4850,5927,2802
%N A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.
%e A380343 The a(5) = 1 through a(17) = 8 partitions (A=10, C=12):
%e A380343   32  .  421  .  54  .  83   .  76    95    843   .  98
%e A380343                         632     742   653   852      863
%e A380343                         641     7321  A31   861      962
%e A380343                                       5432  6432     C32
%e A380343                                       6521  8421     7631
%e A380343                                                      9431
%e A380343                                                      9521
%e A380343                                                      65321
%t A380343 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@#,n+1]&]],{n,0,30}]
%Y A380343 The non-strict version is A379320, ranked by A380217 = A379319/2.
%Y A380343 For n instead of n+1 we have A379733, non-strict A057568.
%Y A380343 The case of equality for non-strict partitions is A380218.
%Y A380343 A000041 counts integer partitions, strict A000009.
%Y A380343 A379666 counts partitions by sum and product.
%Y A380343 A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
%Y A380343 Counting and ranking multisets by comparing sum and product:
%Y A380343 - same: A001055, ranks A301987
%Y A380343 - multiple: A057567, ranks A326155
%Y A380343 - divisor: A057568, ranks A326149
%Y A380343 - greater than: A096276 shifted right, ranks A325038
%Y A380343 - greater or equal: A096276, ranks A325044
%Y A380343 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380343 - less or equal: A319005, ranks A379721, see A025147
%Y A380343 - different: A379736, ranks A379722, see A111133
%Y A380343 Cf. A003963, A069016, A319000, A319916, A325042, A326152, A326156, A379671, A379734.
%K A380343 nonn
%O A380343 0,12
%A A380343 _Gus Wiseman_, Jan 22 2025