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Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).
5

%I #7 Oct 10 2023 09:31:07

%S 1,2,3,5,6,7,8,10,11,13,14,15,16,17,18,19,20,21,22,23,24,26,27,28,29,

%T 31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,50,51,52,53,54,55,

%U 56,57,58,59,60,61,62,64,65,66,67,68,69,71,72,73,74,75,76

%N Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

%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 Also Heinz numbers of integer partitions containing n/2, where n is the sum of all parts.

%e The prime indices of 90 are {1,2,2,3}, with sum 8 and twice maximum 6, so 90 is in the sequence.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Max[prix[#]]!=Total[prix[#]]/2&]

%Y Partitions of this type are counted by A086543.

%Y For length instead of maximum we have the complement of A340387.

%Y The complement is A344415, counted by A035363.

%Y A001221 counts distinct prime factors, A001222 with multiplicity.

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

%Y A334201 adds up all prime indices except the greatest.

%Y A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.

%Y A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.

%Y Cf. A001414, A100959, A238628, A300061, A301987, A316413, A320924, A344414, A344416, A366318.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 10 2023