cp's OEIS Frontend

A380219 Number of integer partitions of n whose product is a proper multiple of n.

%I A380219 #8 Jan 22 2025 06:39:49
%S A380219 0,0,0,0,0,0,0,3,3,3,0,18,0,9,21,75,0,109,0,146,83,43,0,730,224,82,
%T A380219 806,722,0,1782,0,4254,733,258,1923,9558,0,435,1875,16395,0,14625,0,
%U A380219 9857,33053,1150,0,102070,19391,57326,10157,30702,0,207699,47925,200645
%N A380219 Number of integer partitions of n whose product is a proper multiple of n.
%F A380219 a(n) = A057568(n) - A001055(n).
%e A380219 The partition y = (4,3,3,2) has product 72, which is a multiple of 12, so y is counted under a(12).
%e A380219 The a(8) = 3 through a(14) = 9 partitions:
%e A380219   (44)    (63)    (532)   .  (66)       .  (743)
%e A380219   (422)   (333)   (541)      (543)         (752)
%e A380219   (2222)  (3321)  (5221)     (642)         (761)
%e A380219                              (831)         (7322)
%e A380219                              (4332)        (7421)
%e A380219                              (4431)        (72221)
%e A380219                              (5322)        (73211)
%e A380219                              (6222)        (74111)
%e A380219                              (6321)        (722111)
%e A380219                              (6411)
%e A380219                              (33222)
%e A380219                              (43221)
%e A380219                              (43311)
%e A380219                              (62211)
%e A380219                              (322221)
%e A380219                              (332211)
%e A380219                              (432111)
%e A380219                              (3222111)
%t A380219 Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n]&&UnsameQ[Times@@#,n]&]],{n,30}]
%o A380219 (PARI) a(n) = my(nb=0); forpart(p=n, my(vp=vecprod(Vec(p))); if (!(vp%n) && (vp>n), nb++)); nb; \\ _Michel Marcus_, Jan 22 2025
%Y A380219 The non-proper version is A057568, case of equality A001055.
%Y A380219 The case of strict partitions is A379733 - 1.
%Y A380219 The case of partitions without 1's is A379734 - 1.
%Y A380219 These partitions are ranked by A380216.
%Y A380219 A000041 counts integer partitions, strict A000009.
%Y A380219 A379666 counts partitions by sum and product.
%Y A380219 Counting and ranking multisets by comparing sum and product:
%Y A380219 - same: A001055, ranks A301987
%Y A380219 - multiple: A057567, ranks A326155
%Y A380219 - divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
%Y A380219 - greater than: A096276 shifted right, ranks A325038
%Y A380219 - greater or equal: A096276, ranks A325044
%Y A380219 - less than: A114324, ranks A325037, see A318029, A379720
%Y A380219 - less or equal: A319005, ranks A379721, see A025147
%Y A380219 - different: A379736, ranks A379722, see A111133
%Y A380219 Cf. A003963, A028422, A318950, A319000, A319916, A326152, A326156, A379320, A379671, A379735, A380218, A380221.
%K A380219 nonn
%O A380219 1,8
%A A380219 _Gus Wiseman_, Jan 21 2025