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%I #17 Jun 26 2021 12:11:42
%S 5,10,15,25,30,45,50,75,90,100,125,135,150,225,250,270,300,375,405,
%T 450,500,625,675,750,810,900,1000,1125,1215,1250,1350,1500,1875,2025,
%U 2250,2430,2500,2700,3000,3125,3375,3645,3750,4050,4500,5000,5625,6075,6250
%N 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).
%C A number is d-smooth iff its prime divisors are all <= d.
%C A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).
%F Intersection of A080193 and A344291.
%e The sequence of terms together with their prime indices begins:
%e 5: {3} 270: {1,2,2,2,3}
%e 10: {1,3} 300: {1,1,2,3,3}
%e 15: {2,3} 375: {2,3,3,3}
%e 25: {3,3} 405: {2,2,2,2,3}
%e 30: {1,2,3} 450: {1,2,2,3,3}
%e 45: {2,2,3} 500: {1,1,3,3,3}
%e 50: {1,3,3} 625: {3,3,3,3}
%e 75: {2,3,3} 675: {2,2,2,3,3}
%e 90: {1,2,2,3} 750: {1,2,3,3,3}
%e 100: {1,1,3,3} 810: {1,2,2,2,2,3}
%e 125: {3,3,3} 900: {1,1,2,2,3,3}
%e 135: {2,2,2,3} 1000: {1,1,1,3,3,3}
%e 150: {1,2,3,3} 1125: {2,2,3,3,3}
%e 225: {2,2,3,3} 1215: {2,2,2,2,2,3}
%e 250: {1,3,3,3} 1250: {1,3,3,3,3}
%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 A080193, counted by A069905.
%Y The partitions with these Heinz numbers are counted by A325691.
%Y Relaxing the smoothness conditions gives A344291, counted by A110618.
%Y Allowing 3-smoothness gives A344293, counted by A266755.
%Y A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
%Y A035363 counts partitions of n whose length is n/2, ranked by A340387.
%Y A051037 lists 5-smooth numbers (complement: A279622).
%Y A056239 adds up prime indices, row sums of A112798.
%Y A257993 gives the least gap of the partition with Heinz number n.
%Y A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
%Y A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
%Y Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.
%K nonn
%O 1,1
%A _Gus Wiseman_, May 16 2021
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