cp's OEIS Frontend

A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).

%I A344292 #5 May 23 2021 02:59:08
%S A344292 1,3,4,9,10,12,16,27,28,30,36,40,48,64,81,84,88,90,100,108,112,120,
%T A344292 144,160,192,208,243,252,256,264,270,280,300,324,336,352,360,400,432,
%U A344292 448,480,544,576,624,640,729,756,768,784,792,810,832,840,880,900,972
%N A344292 Numbers m whose sum of prime indices A056239(m) is even and is at most twice the number of prime factors counted with multiplicity A001222(m).
%C A344292 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 A344292 Also Heinz numbers of integer partitions of even numbers m with at least m/2 parts, counted by A000070 riffled with 0's, or A025065 with odd positions zeroed out.
%F A344292 Members m of A300061 such that A056239(m) <= 2*A001222(m).
%e A344292 The sequence of terms together with their prime indices begins:
%e A344292        1: {}                 84: {1,1,2,4}
%e A344292        3: {2}                88: {1,1,1,5}
%e A344292        4: {1,1}              90: {1,2,2,3}
%e A344292        9: {2,2}             100: {1,1,3,3}
%e A344292       10: {1,3}             108: {1,1,2,2,2}
%e A344292       12: {1,1,2}           112: {1,1,1,1,4}
%e A344292       16: {1,1,1,1}         120: {1,1,1,2,3}
%e A344292       27: {2,2,2}           144: {1,1,1,1,2,2}
%e A344292       28: {1,1,4}           160: {1,1,1,1,1,3}
%e A344292       30: {1,2,3}           192: {1,1,1,1,1,1,2}
%e A344292       36: {1,1,2,2}         208: {1,1,1,1,6}
%e A344292       40: {1,1,1,3}         243: {2,2,2,2,2}
%e A344292       48: {1,1,1,1,2}       252: {1,1,2,2,4}
%e A344292       64: {1,1,1,1,1,1}     256: {1,1,1,1,1,1,1,1}
%e A344292       81: {2,2,2,2}         264: {1,1,1,2,5}
%t A344292 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A344292 Select[Range[100],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]>=Total[primeMS[#]]/2&]
%Y A344292 These are the Heinz numbers of partitions counted by A000070 and A025065.
%Y A344292 A subset of A300061 (sum of prime indices is even).
%Y A344292 The conjugate opposite version is A320924, counted by A209816.
%Y A344292 The conjugate opposite version allowing odds is A322109, counted by A110618.
%Y A344292 The case of equality is A340387, counted by A000041.
%Y A344292 The opposite version allowing odd weights is A344291, counted by A110618.
%Y A344292 Allowing odd weights gives A344296, counted by A025065.
%Y A344292 The opposite version is A344413, counted by A209816.
%Y A344292 The conjugate version allowing odd weights is A344414, counted by A025065.
%Y A344292 The case of equality in the conjugate case is A344415, counted by A035363.
%Y A344292 The conjugate version is A344416, counted by A000070.
%Y A344292 A001222 counts prime factors with multiplicity.
%Y A344292 A027187 counts partitions of even length, ranked by A028260.
%Y A344292 A056239 adds up prime indices, row sums of A112798.
%Y A344292 A058696 counts partitions of even numbers, ranked by A300061.
%Y A344292 A301987 lists numbers whose sum of prime indices equals their product.
%Y A344292 A330950 counts partitions of n with Heinz number divisible by n.
%Y A344292 A334201 adds up all prime indices except the greatest.
%Y A344292 Cf. A001414, A067538, A316413, A316428, A325037, A325038, A325044, A338914, A344294, A344297.
%K A344292 nonn
%O A344292 1,2
%A A344292 _Gus Wiseman_, May 22 2021