A366321 - OEIS

%I #10 Oct 14 2023 23:55:09

%S 1,3,7,10,13,16,19,21,22,27,28,29,34,36,37,39,43,46,48,52,53,55,57,61,

%T 62,64,66,71,75,76,79,81,82,85,87,88,89,90,91,94,100,101,102,107,108,

%U 111,113,115,116,117,118,120,129,130,131,133,134,136,138,139,144

%N Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

%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.

%e The prime indices of 84 are y = {1,1,2,4}, with even sum 8; but 8/2 = 4 is in y, so 84 is not in the sequence.

%e The terms together with their prime indices begin:

%e     1: {}

%e     3: {2}

%e     7: {4}

%e    10: {1,3}

%e    13: {6}

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

%e    19: {8}

%e    21: {2,4}

%e    22: {1,5}

%e    27: {2,2,2}

%e    28: {1,1,4}

%e    29: {10}

%e    34: {1,7}

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

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

%t Select[Range[100],EvenQ[Total[prix[#]]]&&FreeQ[prix[#],Total[prix[#]]/2]&]

%Y Partitions of this type are counted by A182616, strict A365828.

%Y A066207 lists numbers with all even prime indices, odd A066208.

%Y A086543 lists numbers with at least one odd prime index, counted by A366322.

%Y A300063 ranks partitions of odd numbers.

%Y A366319 ranks partitions of n not containing n/2.

%Y A366321 ranks partitions of 2k that do not contain k.

%Y Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.

%K nonn

%O 0,2

%A _Gus Wiseman_, Oct 13 2023