cp's OEIS Frontend

A379320 Number of integer partitions of n whose product is a multiple of n + 1.

%I A379320 #13 Jan 21 2025 09:53:52
%S A379320 1,0,0,0,0,1,0,2,2,3,0,14,0,7,15,53,0,81,0,110,61,32,0,562,170,62,621,
%T A379320 560,0,1400,0,3387,569,199,1515,7734,0,339,1486,13374,0,11926,0,8033,
%U A379320 27164,913,0,85326,15947,47588,8294,25430,0,174779,39748,169009
%N A379320 Number of integer partitions of n whose product is a multiple of n + 1.
%C A379320 Also the number of integer partitions of n containing 1 whose product is a multiple of n. Without requiring a 1 we get A057568.
%e A379320 The a(5) = 1 through a(11) = 14 partitions:
%e A379320   (3,2)  .  (4,2,1)    (3,3,2)    (5,4)      .  (8,3)
%e A379320             (2,2,2,1)  (3,3,1,1)  (5,2,2)       (4,4,3)
%e A379320                                   (5,2,1,1)     (6,3,2)
%e A379320                                                 (6,4,1)
%e A379320                                                 (4,3,2,2)
%e A379320                                                 (4,3,3,1)
%e A379320                                                 (6,2,2,1)
%e A379320                                                 (3,2,2,2,2)
%e A379320                                                 (3,3,2,2,1)
%e A379320                                                 (4,3,2,1,1)
%e A379320                                                 (6,2,1,1,1)
%e A379320                                                 (3,2,2,2,1,1)
%e A379320                                                 (4,3,1,1,1,1)
%e A379320                                                 (3,2,2,1,1,1,1)
%t A379320 Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n+1]&]],{n,0,30}]
%o A379320 (PARI) a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ _Michel Marcus_, Jan 21 2025
%Y A379320 For n instead of n+1 we have A057568 (strict A379733), ranks A326149.
%Y A379320 These partitions are ranked by A380217 = A379319/2 = (even case of A326149)/2.
%Y A379320 The case of equality is A380218, see also A028422 = A001055 - 1 (ranks A325041).
%Y A379320 A000041 counts integer partitions, strict A000009.
%Y A379320 A379666 counts partitions by sum and product.
%Y A379320 A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
%Y A379320 Counting and ranking multisets by comparing sum and product:
%Y A379320 - same: A001055, ranks A301987
%Y A379320 - multiple: A057567, ranks A326155
%Y A379320 - divisor: A057568, ranks A326149
%Y A379320 - greater than: A096276 shifted right, ranks A325038
%Y A379320 - greater or equal: A096276, ranks A325044
%Y A379320 - less than: A114324, ranks A325037, see A318029, A379720
%Y A379320 - less or equal: A319005, ranks A379721, see A025147
%Y A379320 - different: A379736, ranks A379722, see A111133
%Y A379320 Cf. A003963, A036844, A069016, A318950, A319000, A319916, A324851, A325042, A326150, A326152, A326156, A379671, A379678, A379734.
%K A379320 nonn
%O A379320 0,8
%A A379320 _Gus Wiseman_, Jan 18 2025