cp's OEIS Frontend

A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).

%I A367224 #8 Nov 15 2023 08:22:53
%S A367224 1,2,4,6,8,9,12,15,16,18,20,21,24,30,32,33,36,39,40,42,45,48,50,51,54,
%T A367224 56,57,60,64,66,69,70,72,75,78,80,81,84,87,90,93,96,100,102,105,108,
%U A367224 110,111,112,114,120,123,125,126,128,129,130,132,135,138,140,141
%N A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).
%C A367224 Also numbers m whose prime indices have a submultiset summing to bigomega(m).
%C A367224 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 A367224 These are the Heinz numbers of the partitions counted by A367212.
%e A367224 The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is in the sequence.
%e A367224 The terms together with their prime indices begin:
%e A367224     1: {}
%e A367224     2: {1}
%e A367224     4: {1,1}
%e A367224     6: {1,2}
%e A367224     8: {1,1,1}
%e A367224     9: {2,2}
%e A367224    12: {1,1,2}
%e A367224    15: {2,3}
%e A367224    16: {1,1,1,1}
%e A367224    18: {1,2,2}
%e A367224    20: {1,1,3}
%e A367224    21: {2,4}
%e A367224    24: {1,1,1,2}
%e A367224    30: {1,2,3}
%e A367224    32: {1,1,1,1,1}
%t A367224 prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p], {k}]]]];
%t A367224 Select[Range[100], MemberQ[Total/@prix/@Divisors[#], PrimeOmega[#]]&]
%Y A367224 The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
%Y A367224                sum-full   sum-free   comb-full  comb-free
%Y A367224               -------------------------------------------
%Y A367224   partitions:  A367212    A367213    A367218    A367219
%Y A367224   strict:      A367214    A367215    A367220    A367221
%Y A367224   subsets:     A367216    A367217    A367222    A367223
%Y A367224   ranks:       A367224*   A367225    A367226    A367227
%Y A367224 A000700 counts self-conjugate partitions, ranks A088902.
%Y A367224 A002865 counts partitions whose length is a part, ranks A325761.
%Y A367224 A005117 ranks strict integer partitions, counted by A000009.
%Y A367224 A066208 ranks partitions into odd parts, also counted by A000009.
%Y A367224 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A367224 A126796 counts complete partitions, ranks A325781.
%Y A367224 A229816 counts partitions whose length is not a part, ranks A367107.
%Y A367224 A237668 counts sum-full partitions, ranks A364532.
%Y A367224 Triangles:
%Y A367224 A046663 counts partitions of n without a subset-sum k, strict A365663.
%Y A367224 A365543 counts partitions of n with a subset-sum k, strict A365661.
%Y A367224 A365658 counts partitions by number of subset-sums, strict A365832.
%Y A367224 Cf. A000720, A055396, A061395, A106529, A299729, A304792, A363225, A365830.
%K A367224 nonn
%O A367224 1,2
%A A367224 _Gus Wiseman_, Nov 14 2023