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Heinz numbers of integer partitions containing at least one odd part. Numbers divisible by at least one prime of odd index.
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%I #5 Oct 14 2023 23:52:49

%S 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 34,35,36,38,40,41,42,44,45,46,47,48,50,51,52,54,55,56,58,59,60,62,64,

%U 65,66,67,68,69,70,72,73,74,75,76,77,78,80,82,83,84,85,86

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

%C 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 A257991(a(n)) > 0.

%e The terms together with their prime indices begin:

%e 2: {1}

%e 4: {1,1}

%e 5: {3}

%e 6: {1,2}

%e 8: {1,1,1}

%e 10: {1,3}

%e 11: {5}

%e 12: {1,1,2}

%e 14: {1,4}

%e 15: {2,3}

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

%e 17: {7}

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 22: {1,5}

%e 23: {9}

%e 24: {1,1,1,2}

%t Select[Range[100],Or@@OddQ/@PrimePi/@First/@FactorInteger[#]&]

%Y The complement is A066207, counted by A035363.

%Y For all odd parts we have A066208, counted by A000009.

%Y Partitions of this type are counted by A086543.

%Y For even instead of odd we have A324929, counted by A047967.

%Y A031368 lists primes of odd index.

%Y A112798 list prime indices, sum A056239.

%Y A257991 counts odd prime indices, distinct A324966.

%Y Cf. A000720, A001222, A003963, A257992, A318400, A324927, A358137.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 14 2023