cp's OEIS Frontend

A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.

%I A366321 #10 Oct 14 2023 23:55:09
%S A366321 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 A366321 62,64,66,71,75,76,79,81,82,85,87,88,89,90,91,94,100,101,102,107,108,
%U A366321 111,113,115,116,117,118,120,129,130,131,133,134,136,138,139,144
%N A366321 Numbers m whose prime indices have even sum k such that k/2 is not a prime index of m.
%C A366321 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 A366321 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 A366321 The terms together with their prime indices begin:
%e A366321     1: {}
%e A366321     3: {2}
%e A366321     7: {4}
%e A366321    10: {1,3}
%e A366321    13: {6}
%e A366321    16: {1,1,1,1}
%e A366321    19: {8}
%e A366321    21: {2,4}
%e A366321    22: {1,5}
%e A366321    27: {2,2,2}
%e A366321    28: {1,1,4}
%e A366321    29: {10}
%e A366321    34: {1,7}
%e A366321    36: {1,1,2,2}
%t A366321 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366321 Select[Range[100],EvenQ[Total[prix[#]]]&&FreeQ[prix[#],Total[prix[#]]/2]&]
%Y A366321 Partitions of this type are counted by A182616, strict A365828.
%Y A366321 A066207 lists numbers with all even prime indices, odd A066208.
%Y A366321 A086543 lists numbers with at least one odd prime index, counted by A366322.
%Y A366321 A300063 ranks partitions of odd numbers.
%Y A366321 A366319 ranks partitions of n not containing n/2.
%Y A366321 A366321 ranks partitions of 2k that do not contain k.
%Y A366321 Cf. A000041, A006827, A047967, A320924, A339662, A365825, A365920, A366318, A366528, A366530.
%K A366321 nonn
%O A366321 0,2
%A A366321 _Gus Wiseman_, Oct 13 2023