cp's OEIS Frontend

A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.

%I A381806 #18 Mar 28 2025 14:13:20
%S A381806 4,8,9,16,24,25,27,32,40,48,49,54,56,64,72,80,81,88,96,104,108,112,
%T A381806 121,125,128,135,136,144,152,160,162,169,176,184,189,192,200,208,216,
%U A381806 224,232,240,243,248,250,256,272,288,289,296,297,304,320,324,328,336
%N A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.
%C A381806 First differs from A212164 in having 3600.
%C A381806 First differs from A293243 in having 18000.
%C A381806 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A381806 Also numbers whose prime indices cannot be partitioned into a multiset of sets with distinct sums.
%e A381806 There are 4 factorizations of 18000 into squarefree numbers:
%e A381806   (2*2*3*5*10*30)
%e A381806   (2*2*5*6*10*15)
%e A381806   (2*2*10*15*30)
%e A381806   (2*5*6*10*30)
%e A381806 but none of these has all distinct sums of prime indices, so 18000 is in the sequence.
%t A381806 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381806 sqfics[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfics[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
%t A381806 Select[Range[nn],Length[Select[sqfics[#],UnsameQ@@hwt/@#&]]==0&]
%Y A381806 Strongly normal multisets of this type are counted by A292444.
%Y A381806 These are the zeros in A381633, see A050320, A321469, A381078, A381634.
%Y A381806 For distinct blocks see A050326, A293243, A293511, A358914, A381441.
%Y A381806 For more on set multipartitions see A089259, A116540, A270995, A296119, A318360.
%Y A381806 For more on set multipartitions with distinct sums see A279785, A381718.
%Y A381806 For constant instead of strict blocks we have A381636, see A381635, A381716.
%Y A381806 Partitions of this type are counted by A381990, complement A381992.
%Y A381806 The complement is A382075.
%Y A381806 A001055 counts multiset partitions, strict A045778.
%Y A381806 A003963 gives product of prime indices.
%Y A381806 A055396 gives least prime index, greatest A061395.
%Y A381806 A056239 adds up prime indices, row sums of A112798.
%Y A381806 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A381806 Cf. A000688, A000720, A001222, A005117, A299202, A300385, A381454, A381870.
%K A381806 nonn
%O A381806 1,1
%A A381806 _Gus Wiseman_, Mar 12 2025