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A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.

%I A344296 #13 May 20 2021 23:05:33
%S A344296 1,2,3,4,6,8,9,10,12,16,18,20,24,27,28,30,32,36,40,48,54,56,60,64,72,
%T A344296 80,81,84,88,90,96,100,108,112,120,128,144,160,162,168,176,180,192,
%U A344296 200,208,216,224,240,243,252,256,264,270,280,288,300,320,324,336,352
%N A344296 Numbers with at least as many prime factors (counted with multiplicity) as half their sum of prime indices.
%C A344296 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 A344296 These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.
%F A344296 A056239(a(n)) <= 2*A001222(a(n)).
%F A344296 a(n) = A322136(n)/4.
%e A344296 The sequence of terms together with their prime indices begins:
%e A344296       1: {}            30: {1,2,3}
%e A344296       2: {1}           32: {1,1,1,1,1}
%e A344296       3: {2}           36: {1,1,2,2}
%e A344296       4: {1,1}         40: {1,1,1,3}
%e A344296       6: {1,2}         48: {1,1,1,1,2}
%e A344296       8: {1,1,1}       54: {1,2,2,2}
%e A344296       9: {2,2}         56: {1,1,1,4}
%e A344296      10: {1,3}         60: {1,1,2,3}
%e A344296      12: {1,1,2}       64: {1,1,1,1,1,1}
%e A344296      16: {1,1,1,1}     72: {1,1,1,2,2}
%e A344296      18: {1,2,2}       80: {1,1,1,1,3}
%e A344296      20: {1,1,3}       81: {2,2,2,2}
%e A344296      24: {1,1,1,2}     84: {1,1,2,4}
%e A344296      27: {2,2,2}       88: {1,1,1,5}
%e A344296      28: {1,1,4}       90: {1,2,2,3}
%t A344296 Select[Range[100],PrimeOmega[#]>=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]
%Y A344296 The case with difference at least 1 is A322136.
%Y A344296 The case of equality is A340387, counted by A000041 or A035363.
%Y A344296 The opposite version is A344291, counted by A110618.
%Y A344296 The conjugate version is A344414, with even-weight case A344416.
%Y A344296 A025065 counts palindromic partitions, ranked by A265640.
%Y A344296 A056239 adds up prime indices, row sums of A112798.
%Y A344296 A300061 lists numbers whose sum of prime indices is even.
%Y A344296 Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.
%K A344296 nonn
%O A344296 1,2
%A A344296 _Gus Wiseman_, May 16 2021