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A338470 Number of integer partitions of n with no part dividing all the others.

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