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A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

%I A344293 #12 May 26 2021 02:30:25
%S A344293 1,3,5,9,10,15,25,27,30,45,50,75,81,90,100,125,135,150,225,243,250,
%T A344293 270,300,375,405,450,500,625,675,729,750,810,900,1000,1125,1215,1250,
%U A344293 1350,1500,1875,2025,2187,2250,2430,2500,2700,3000,3125,3375,3645,3750,4050
%N A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).
%C A344293 A number is 5-smooth if its prime divisors are all <= 5.
%C A344293 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.
%F A344293 Intersection of A051037 and A344291.
%e A344293 The sequence of terms together with their prime indices begins:
%e A344293        1: {}            125: {3,3,3}
%e A344293        3: {2}           135: {2,2,2,3}
%e A344293        5: {3}           150: {1,2,3,3}
%e A344293        9: {2,2}         225: {2,2,3,3}
%e A344293       10: {1,3}         243: {2,2,2,2,2}
%e A344293       15: {2,3}         250: {1,3,3,3}
%e A344293       25: {3,3}         270: {1,2,2,2,3}
%e A344293       27: {2,2,2}       300: {1,1,2,3,3}
%e A344293       30: {1,2,3}       375: {2,3,3,3}
%e A344293       45: {2,2,3}       405: {2,2,2,2,3}
%e A344293       50: {1,3,3}       450: {1,2,2,3,3}
%e A344293       75: {2,3,3}       500: {1,1,3,3,3}
%e A344293       81: {2,2,2,2}     625: {3,3,3,3}
%e A344293       90: {1,2,2,3}     675: {2,2,2,3,3}
%e A344293      100: {1,1,3,3}     729: {2,2,2,2,2,2}
%t A344293 Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]
%Y A344293 Allowing any number of parts and sum gives A051037, counted by A001399.
%Y A344293 These are Heinz numbers of the partitions counted by A266755.
%Y A344293 Allowing parts > 5 gives A344291, counted by A110618.
%Y A344293 The non-3-smooth case is A344294, counted by A325691.
%Y A344293 Requiring the sum of prime indices to be even gives A344295.
%Y A344293 A000070 counts non-multigraphical partitions, ranked by A344292.
%Y A344293 A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
%Y A344293 A035363 counts partitions of n with n/2 parts, ranked by A340387.
%Y A344293 A056239 adds up prime indices, row sums of A112798.
%Y A344293 A300061 ranks partitions of even numbers, with 5-smooth case A344297.
%Y A344293 Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.
%K A344293 nonn
%O A344293 1,2
%A A344293 _Gus Wiseman_, May 16 2021