cp's OEIS Frontend

A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

%I A367397 #7 Nov 21 2023 08:22:02
%S A367397 4,12,18,30,36,40,42,54,60,66,78,81,90,100,102,112,114,120,126,135,
%T A367397 138,140,150,168,174,180,186,189,198,210,220,222,225,234,246,250,252,
%U A367397 258,260,270,280,282,297,300,306,315,318,330,336,340,342,350,351,352,354
%N A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.
%C A367397 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 A367397 These are the Heinz numbers of the partitions counted by A367394.
%t A367397 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A367397 Select[Range[100],MemberQ[Total/@Subsets[prix[#],{2}],PrimeOmega[#]]&]
%Y A367397 The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
%Y A367397              sum-full  sum-free  comb-full comb-free semi-full semi-free
%Y A367397             -----------------------------------------------------------
%Y A367397 partitions:  A367212   A367213   A367218   A367219   A367394   A367398
%Y A367397 strict:      A367214   A367215   A367220   A367221   A367395   A367399
%Y A367397 subsets:     A367216   A367217   A367222   A367223   A367396   A367400
%Y A367397 ranks:       A367224   A367225   A367226   A367227   A367397*  A367401
%Y A367397 A325761 ranks partitions whose length is a part, counted by A002865.
%Y A367397 A088809 and A093971 count subsets containing semi-sums.
%Y A367397 A236912 counts partitions with no semi-sum of the parts, ranks A364461.
%Y A367397 A237113 counts partitions with a semi-sum of the parts, ranks A364462.
%Y A367397 A304792 counts subset-sums of partitions, strict A365925.
%Y A367397 A366738 counts semi-sums of partitions, strict A366741.
%Y A367397 Triangles:
%Y A367397 A365381 counts subsets with a subset summing to k, complement A366320.
%Y A367397 A365541 counts subsets with a semi-sum k.
%Y A367397 A367404 counts partitions with a semi-sum k, strict A367405.
%Y A367397 Cf. A000700, A229816, A237667, A237668, A238628, A363225, A364272, A365543, A365658, A365918, A366740.
%K A367397 nonn
%O A367397 1,1
%A A367397 _Gus Wiseman_, Nov 21 2023