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%I A327778 #17 May 18 2021 06:32:15
%S A327778 0,1,1,1,1,1,2,1,1,1,5,1,11,1,11,23,1,1,23,1,85,85,45,1,152,1,84,1,
%T A327778 451,1,1787,1,1,735,260,1925,1908,1,437,1877,4623,1,14630,1,6934,
%U A327778 10519,1152,1,6791,1,1817,10159,22556,1,2819,47927,69333,22010,4310,1
%N A327778 Number of integer partitions of n whose LCM is a multiple of n.
%H A327778 Alois P. Heinz, <a href="/A327778/b327778.txt">Table of n, a(n) for n = 0..1000</a>
%F A327778 a(n) = 1 <=> n in { A000961 }. - _Alois P. Heinz_, Sep 26 2019
%e A327778 The partitions of n = 6, 10, 12, and 15 whose LCM is a multiple of n:
%e A327778 (6) (10) (12) (15)
%e A327778 (3,2,1) (5,3,2) (5,4,3) (6,5,4)
%e A327778 (5,4,1) (6,4,2) (7,5,3)
%e A327778 (5,2,2,1) (8,3,1) (9,5,1)
%e A327778 (5,2,1,1,1) (4,3,3,2) (10,3,2)
%e A327778 (4,4,3,1) (5,4,3,3)
%e A327778 (6,4,1,1) (5,5,3,2)
%e A327778 (4,3,2,2,1) (6,5,2,2)
%e A327778 (4,3,3,1,1) (6,5,3,1)
%e A327778 (4,3,2,1,1,1) (10,3,1,1)
%e A327778 (4,3,1,1,1,1,1) (5,3,3,2,2)
%e A327778 (5,3,3,3,1)
%e A327778 (5,4,3,2,1)
%e A327778 (5,5,3,1,1)
%e A327778 (6,5,2,1,1)
%e A327778 (5,3,2,2,2,1)
%e A327778 (5,3,3,2,1,1)
%e A327778 (5,4,3,1,1,1)
%e A327778 (6,5,1,1,1,1)
%e A327778 (5,3,2,2,1,1,1)
%e A327778 (5,3,3,1,1,1,1)
%e A327778 (5,3,2,1,1,1,1,1)
%e A327778 (5,3,1,1,1,1,1,1,1)
%p A327778 a:= proc(m) option remember; local b; b:=
%p A327778 proc(n, i, l) option remember; `if`(n=0 or i=1,
%p A327778 `if`(l=m, 1, 0), `if`(i<2, 0, b(n, i-1, l))+
%p A327778 b(n-i, min(n-i, i), igcd(m, ilcm(l, i))))
%p A327778 end; `if`(isprime(m), 1, b(m$2, 1))
%p A327778 end:
%p A327778 seq(a(n), n=0..60); # _Alois P. Heinz_, Sep 26 2019
%t A327778 Table[Length[Select[IntegerPartitions[n],Divisible[LCM@@#,n]&]],{n,30}]
%t A327778 (* Second program: *)
%t A327778 a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] = If[n == 0 || i == 1, If[l == m, 1, 0], If[i<2, 0, b[n, i - 1, l]] + b[n - i, Min[n - i, i], GCD[m, LCM[l, i]]]]; If[PrimeQ[m], 1, b[m, m, 1]]];
%t A327778 a /@ Range[0, 60] (* _Jean-François Alcover_, May 18 2021, after _Alois P. Heinz_ *)
%Y A327778 The Heinz numbers of these partitions are given by A327783.
%Y A327778 Partitions whose LCM is equal to their sum are A074761.
%Y A327778 Partitions whose LCM is greater than their sum are A327779.
%Y A327778 Partitions whose LCM is less than their sum are A327781.
%Y A327778 Cf. A000961, A018818, A067538, A290103, A319333, A326842, A326843, A327780.
%K A327778 nonn
%O A327778 0,7
%A A327778 _Gus Wiseman_, Sep 25 2019