%I #20 May 09 2021 12:45:18
%S 1,0,0,0,0,1,0,3,2,5,5,13,7,23,21,33,35,65,55,104,97,151,166,252,235,
%T 377,399,549,591,846,858,1237,1311,1749,1934,2556,2705,3659,3991,5090,
%U 5608,7244,7841,10086,11075,13794,15420,19195,21003,26240,29089,35483
%N Number of integer partitions of n with no part dividing all the others.
%C Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.
%H Andrew Howroyd, <a href="/A338470/b338470.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - _Andrew Howroyd_, Mar 25 2021
%e The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
%e (32) . (43) (53) (54) (64) (65) (75)
%e (52) (332) (72) (73) (74) (543)
%e (322) (432) (433) (83) (552)
%e (522) (532) (92) (732)
%e (3222) (3322) (443) (4332)
%e (533) (5322)
%e (542) (33222)
%e (632)
%e (722)
%e (3332)
%e (4322)
%e (5222)
%e (32222)
%t Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
%t (* Second program: *)
%t a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
%t a /@ Range[0, 50] (* _Jean-François Alcover_, May 09 2021, after _Andrew Howroyd_ *)
%o (PARI) a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ _Andrew Howroyd_, Mar 25 2021
%Y The complement is A083710 (strict: A097986).
%Y The strict case is A341450.
%Y The Heinz numbers of these partitions are A342193.
%Y The dual version is A343341.
%Y The case with maximum part not divisible by all the others is A343342.
%Y The case with maximum part divisible by all the others is A343344.
%Y A000005 counts divisors.
%Y A000041 counts partitions.
%Y A000070 counts partitions with a selected part.
%Y A001787 count normal multisets with a selected position.
%Y A006128 counts partitions with a selected position.
%Y A015723 counts strict partitions with a selected part.
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A276024 counts positive subset sums.
%Y Sequences with similar formulas: A024994, A047966, A047968, A168111.
%Y Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.
%K nonn
%O 0,8
%A _Gus Wiseman_, Mar 23 2021