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A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.

%I A366322 #5 Oct 14 2023 23:52:49
%S A366322 2,4,5,6,8,10,11,12,14,15,16,17,18,20,22,23,24,25,26,28,30,31,32,33,
%T A366322 34,35,36,38,40,41,42,44,45,46,47,48,50,51,52,54,55,56,58,59,60,62,64,
%U A366322 65,66,67,68,69,70,72,73,74,75,76,77,78,80,82,83,84,85,86
%N A366322 Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.
%C A366322 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%F A366322 A257991(a(n)) > 0.
%e A366322 The terms together with their prime indices begin:
%e A366322     2: {1}
%e A366322     4: {1,1}
%e A366322     5: {3}
%e A366322     6: {1,2}
%e A366322     8: {1,1,1}
%e A366322    10: {1,3}
%e A366322    11: {5}
%e A366322    12: {1,1,2}
%e A366322    14: {1,4}
%e A366322    15: {2,3}
%e A366322    16: {1,1,1,1}
%e A366322    17: {7}
%e A366322    18: {1,2,2}
%e A366322    20: {1,1,3}
%e A366322    22: {1,5}
%e A366322    23: {9}
%e A366322    24: {1,1,1,2}
%t A366322 Select[Range[100],Or@@OddQ/@PrimePi/@First/@FactorInteger[#]&]
%Y A366322 The complement is A066207, counted by A035363.
%Y A366322 For all odd parts we have A066208, counted by A000009.
%Y A366322 Partitions of this type are counted by A086543.
%Y A366322 For even instead of odd we have A324929, counted by A047967.
%Y A366322 A031368 lists primes of odd index.
%Y A366322 A112798 list prime indices, sum A056239.
%Y A366322 A257991 counts odd prime indices, distinct A324966.
%Y A366322 Cf. A000720, A001222, A003963, A257992, A318400, A324927, A358137.
%K A366322 nonn
%O A366322 1,1
%A A366322 _Gus Wiseman_, Oct 14 2023