A367397 - OEIS

A367397

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

%I #7 Nov 21 2023 08:22:02

%S 4,12,18,30,36,40,42,54,60,66,78,81,90,100,102,112,114,120,126,135,

%T 138,140,150,168,174,180,186,189,198,210,220,222,225,234,246,250,252,

%U 258,260,270,280,282,297,300,306,315,318,330,336,340,342,350,351,352,354

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

%C 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 These are the Heinz numbers of the partitions counted by A367394.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],MemberQ[Total/@Subsets[prix[#],{2}],PrimeOmega[#]]&]

%Y 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 sum-full sum-free comb-full comb-free semi-full semi-free

%Y -----------------------------------------------------------

%Y partitions: A367212 A367213 A367218 A367219 A367394 A367398

%Y strict: A367214 A367215 A367220 A367221 A367395 A367399

%Y subsets: A367216 A367217 A367222 A367223 A367396 A367400

%Y ranks: A367224 A367225 A367226 A367227 A367397* A367401

%Y A325761 ranks partitions whose length is a part, counted by A002865.

%Y A088809 and A093971 count subsets containing semi-sums.

%Y A236912 counts partitions with no semi-sum of the parts, ranks A364461.

%Y A237113 counts partitions with a semi-sum of the parts, ranks A364462.

%Y A304792 counts subset-sums of partitions, strict A365925.

%Y A366738 counts semi-sums of partitions, strict A366741.

%Y Triangles:

%Y A365381 counts subsets with a subset summing to k, complement A366320.

%Y A365541 counts subsets with a semi-sum k.

%Y A367404 counts partitions with a semi-sum k, strict A367405.

%Y Cf. A000700, A229816, A237667, A237668, A238628, A363225, A364272, A365543, A365658, A365918, A366740.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 21 2023