cp's OEIS Frontend

A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).

%I A367226 #7 Nov 15 2023 08:06:38
%S A367226 1,2,4,6,8,9,10,12,14,15,16,18,20,21,22,24,26,28,30,32,33,34,36,38,39,
%T A367226 40,42,44,45,46,48,50,51,52,54,56,57,58,60,62,64,66,68,69,70,72,74,75,
%U A367226 76,78,80,81,82,84,86,87,88,90,92,93,94,96,98,100,102,104
%N A367226 Numbers m whose prime indices have a nonnegative linear combination equal to bigomega(m).
%C A367226 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 A367226 These are the Heinz numbers of the partitions counted by A367218.
%e A367226 The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is in the sequence.
%e A367226 The terms together with their prime indices begin:
%e A367226     1: {}
%e A367226     2: {1}
%e A367226     4: {1,1}
%e A367226     6: {1,2}
%e A367226     8: {1,1,1}
%e A367226     9: {2,2}
%e A367226    10: {1,3}
%e A367226    12: {1,1,2}
%e A367226    14: {1,4}
%e A367226    15: {2,3}
%e A367226    16: {1,1,1,1}
%e A367226    18: {1,2,2}
%e A367226    20: {1,1,3}
%e A367226    21: {2,4}
%e A367226    22: {1,5}
%e A367226    24: {1,1,1,2}
%e A367226    26: {1,6}
%e A367226    28: {1,1,4}
%e A367226    30: {1,2,3}
%e A367226    32: {1,1,1,1,1}
%t A367226 prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A367226 combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t A367226 Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]!={}&]
%Y A367226 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 A367226                sum-full   sum-free   comb-full  comb-free
%Y A367226               -------------------------------------------
%Y A367226   partitions:  A367212    A367213    A367218    A367219
%Y A367226   strict:      A367214    A367215    A367220    A367221
%Y A367226   subsets:     A367216    A367217    A367222    A367223
%Y A367226   ranks:       A367224    A367225    A367226*   A367227
%Y A367226 A000700 counts self-conjugate partitions, ranks A088902.
%Y A367226 A002865 counts partitions whose length is a part, ranks A325761.
%Y A367226 A005117 ranks strict partitions, counted by A000009.
%Y A367226 A046663 counts partitions of n without a subset-sum k, strict A365663.
%Y A367226 A066208 ranks partitions into odd parts, counted by A000009.
%Y A367226 A088809/A093971/A364534 count certain types of sum-full subsets.
%Y A367226 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A367226 A126796 counts complete partitions, ranks A325781.
%Y A367226 A237668 counts sum-full partitions, ranks A364532.
%Y A367226 A365046 counts combination-full subsets, differences of A364914.
%Y A367226 Cf. A000720, A088314, A106529, A116861, A237113, A238628, A299702, A364347, A365073, A367107.
%K A367226 nonn
%O A367226 1,2
%A A367226 _Gus Wiseman_, Nov 15 2023