%I #10 Apr 19 2021 08:05:39

%S 1,1,1,2,2,2,3,3,3,4,4,4,6,5,4,6,6,6,8,7,7,10,9,9,12,10,8,11,11,10,14,

%T 13,11,13,12,15,20,17,15,19,19,19,22,18,17,23,22,22,28,25,24,31,28,26,

%U 32,32,30,34,32,29,37,33,27,36,33,34,44,38,36,45,45

%N Number of strict integer partitions of n with a part divisible by all the others.

%C Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

%H Andrew Howroyd, <a href="/A343347/b343347.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - _Andrew Howroyd_, Apr 17 2021

%e The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):

%e 1 2 3 4 5 6 7 8 9 A B C D E F

%e 21 31 41 42 61 62 63 82 A1 84 C1 C2 A5

%e 51 421 71 81 91 632 93 841 D1 C3

%e 621 631 821 A2 931 842 E1

%e B1 A21 C21

%e 6321 8421

%t Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

%o (PARI) seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ _Andrew Howroyd_, Apr 17 2021

%Y The dual version is A097986 (non-strict: A083710).

%Y The non-strict version is A130689 (Heinz numbers: complement of A343337).

%Y The strict complement is counted by A343377.

%Y The case with smallest part divisible by all the others is A343378.

%Y The case with smallest part not divisible by all the others is A343380.

%Y A000005 counts divisors.

%Y A000009 counts strict partitions.

%Y A000070 counts partitions with a selected part.

%Y A015723 counts strict partitions with a selected part.

%Y A018818 counts partitions into divisors (strict: A033630).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y A339564 counts factorizations with a selected factor.

%Y Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

%K nonn

%O 0,4

%A _Gus Wiseman_, Apr 16 2021