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