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A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

%I A344294 #17 Jun 26 2021 12:11:42
%S A344294 5,10,15,25,30,45,50,75,90,100,125,135,150,225,250,270,300,375,405,
%T A344294 450,500,625,675,750,810,900,1000,1125,1215,1250,1350,1500,1875,2025,
%U A344294 2250,2430,2500,2700,3000,3125,3375,3645,3750,4050,4500,5000,5625,6075,6250
%N A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).
%C A344294 A number is d-smooth iff its prime divisors are all <= d.
%C A344294 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 A344294 Intersection of A080193 and A344291.
%e A344294 The sequence of terms together with their prime indices begins:
%e A344294        5: {3}           270: {1,2,2,2,3}
%e A344294       10: {1,3}         300: {1,1,2,3,3}
%e A344294       15: {2,3}         375: {2,3,3,3}
%e A344294       25: {3,3}         405: {2,2,2,2,3}
%e A344294       30: {1,2,3}       450: {1,2,2,3,3}
%e A344294       45: {2,2,3}       500: {1,1,3,3,3}
%e A344294       50: {1,3,3}       625: {3,3,3,3}
%e A344294       75: {2,3,3}       675: {2,2,2,3,3}
%e A344294       90: {1,2,2,3}     750: {1,2,3,3,3}
%e A344294      100: {1,1,3,3}     810: {1,2,2,2,2,3}
%e A344294      125: {3,3,3}       900: {1,1,2,2,3,3}
%e A344294      135: {2,2,2,3}    1000: {1,1,1,3,3,3}
%e A344294      150: {1,2,3,3}    1125: {2,2,3,3,3}
%e A344294      225: {2,2,3,3}    1215: {2,2,2,2,2,3}
%e A344294      250: {1,3,3,3}    1250: {1,3,3,3,3}
%t A344294 Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]==5&]
%Y A344294 Allowing any number of parts and sum gives A080193, counted by A069905.
%Y A344294 The partitions with these Heinz numbers are counted by A325691.
%Y A344294 Relaxing the smoothness conditions gives A344291, counted by A110618.
%Y A344294 Allowing 3-smoothness gives A344293, counted by A266755.
%Y A344294 A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
%Y A344294 A035363 counts partitions of n whose length is n/2, ranked by A340387.
%Y A344294 A051037 lists 5-smooth numbers (complement: A279622).
%Y A344294 A056239 adds up prime indices, row sums of A112798.
%Y A344294 A257993 gives the least gap of the partition with Heinz number n.
%Y A344294 A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
%Y A344294 A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
%Y A344294 Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.
%K A344294 nonn
%O A344294 1,1
%A A344294 _Gus Wiseman_, May 16 2021