A344296 - OEIS

%I #13 May 20 2021 23:05:33

%S 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 80,81,84,88,90,96,100,108,112,120,128,144,160,162,168,176,180,192,

%U 200,208,216,224,240,243,252,256,264,270,280,288,300,320,324,336,352

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

%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.

%C These are the Heinz numbers of certain partitions counted by A025065, but different from palindromic partitions, which have Heinz numbers A265640.

%F A056239(a(n)) <= 2*A001222(a(n)).

%F a(n) = A322136(n)/4.

%e The sequence of terms together with their prime indices begins:

%e       1: {}            30: {1,2,3}

%e       2: {1}           32: {1,1,1,1,1}

%e       3: {2}           36: {1,1,2,2}

%e       4: {1,1}         40: {1,1,1,3}

%e       6: {1,2}         48: {1,1,1,1,2}

%e       8: {1,1,1}       54: {1,2,2,2}

%e       9: {2,2}         56: {1,1,1,4}

%e      10: {1,3}         60: {1,1,2,3}

%e      12: {1,1,2}       64: {1,1,1,1,1,1}

%e      16: {1,1,1,1}     72: {1,1,1,2,2}

%e      18: {1,2,2}       80: {1,1,1,1,3}

%e      20: {1,1,3}       81: {2,2,2,2}

%e      24: {1,1,1,2}     84: {1,1,2,4}

%e      27: {2,2,2}       88: {1,1,1,5}

%e      28: {1,1,4}       90: {1,2,2,3}

%t Select[Range[100],PrimeOmega[#]>=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]

%Y The case with difference at least 1 is A322136.

%Y The case of equality is A340387, counted by A000041 or A035363.

%Y The opposite version is A344291, counted by A110618.

%Y The conjugate version is A344414, with even-weight case A344416.

%Y A025065 counts palindromic partitions, ranked by A265640.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A300061 lists numbers whose sum of prime indices is even.

%Y Cf. A001399, A002865, A025147, A027336, A036036, A067712, A244990, A261144, A325691, A344293, A344295.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 16 2021