cp's OEIS Frontend

A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

%I A380221 #6 Jan 23 2025 00:18:52
%S A380221 1,0,0,0,0,1,0,0,0,1,0,2,0,2,3,3,0,4,0,9,6,4,0,22,5,6,15,28,0,54,0,49,
%T A380221 30,14,57,134,0,22,58,219,0,242,0,180,349,44,0,722,113,369,196,404,0,
%U A380221 994,556,1363,338,111,0,3016,0,150,2569,3150,1485,2815,0
%N A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.
%C A380221 Also the number of strict integer partitions of n - 1 not containing 1 whose product of parts is a multiple of n. These are strict integer factorizations of multiples of n summing to n - 1.
%e A380221 The a(6) = 1 through a(16) = 3 partitions:
%e A380221   (3,2,1) . . . (5,4,1) . (8,3,1)   . (7,6,1)   (9,5,1)     (8,4,3,1)
%e A380221                           (6,3,2,1)   (7,4,2,1) (6,5,3,1)   (8,5,2,1)
%e A380221                                                 (5,4,3,2,1) (6,4,3,2,1)
%t A380221 Table[Length[Select[IntegerPartitions[n],MemberQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]
%Y A380221 Positions of 0 after 9 appear to be the prime numbers A000040.
%Y A380221 The non-strict version is A379320 shifted right, ranks A380217 = A379319/2.
%Y A380221 Not requiring 1 gives A379733.
%Y A380221 For n instead of n+1 we have A379735 shifted left, non-strict A379734.
%Y A380221 Partitions of this type are ranked by A379845.
%Y A380221 The case of equality for non-strict partitions is A380218 shifted left.
%Y A380221 A000041 counts integer partitions, strict A000009.
%Y A380221 A379666 counts partitions by sum and product.
%Y A380221 A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
%Y A380221 Counting and ranking multisets by comparing sum and product:
%Y A380221 - same: A001055, ranks A301987
%Y A380221 - multiple: A057567, ranks A326155
%Y A380221 - divisor: A057568, ranks A326149
%Y A380221 - greater than: A096276 shifted right, ranks A325038
%Y A380221 - greater or equal: A096276, ranks A325044
%Y A380221 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380221 - less or equal: A319005, ranks A379721, see A025147
%Y A380221 - different: A379736, ranks A379722, see A111133
%Y A380221 Cf. A003963, A069016, A319000, A319916, A325042, A326150, A326152, A326156, A379671, A379678.
%K A380221 nonn
%O A380221 1,12
%A A380221 _Gus Wiseman_, Jan 22 2025