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%I #9 Jan 07 2025 19:05:13
%S 1,1,1,1,1,2,1,1,2,3,1,5,1,5,7,7,1,12,1,20,15,11,1,48,12,16,33,61,1,
%T 121,1,105,67,34,126,292,1,49,128,471,1,522,1,387,751,96,1,1556,246,
%U 792,422,869,1,2126,1191,2904,726,240,1,6393,1,321,5460,6711
%N Number of strict integer partitions of n whose product of parts is a multiple of n.
%C Partitions of this type are ranked by the squarefree terms of A326149.
%e The a(n) partitions for n = 1, 6, 10, 12, 15, 18:
%e (1) (6) (10) (12) (15) (18)
%e (3,2,1) (5,3,2) (5,4,3) (6,5,4) (12,6)
%e (5,4,1) (6,4,2) (7,5,3) (9,5,4)
%e (8,3,1) (9,5,1) (9,6,3)
%e (6,3,2,1) (10,3,2) (9,7,2)
%e (6,5,3,1) (9,8,1)
%e (5,4,3,2,1) (6,5,4,3)
%e (7,6,3,2)
%e (8,6,3,1)
%e (9,4,3,2)
%e (9,6,2,1)
%e (12,3,2,1)
%p b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, `if`(t=1, 1, 0), b(n, i-1, t)+
%p b(n-i, min(i-1, n-i), t/igcd(i, t))))
%p end:
%p a:= n-> `if`(isprime(n), 1, b(n$3)):
%p seq(a(n), n=1..70); # _Alois P. Heinz_, Jan 07 2025
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]
%Y The non-strict opposite version is A057567, ranks A326155.
%Y The non-strict version is A057568, ranks A326149.
%Y The case of partitions without 1's is A379735, non-strict A379734.
%Y A319005 counts partitions with product >= sum, ranks A379721.
%Y A114324 counts partitions with product greater than sum, ranks A325037.
%Y Cf. A001055, A003963, A069016, A096276, A111133, A318950, A319000, A319057, A319916, A326152, A379720.
%K nonn
%O 1,6
%A _Gus Wiseman_, Jan 07 2025