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A300063 Heinz numbers of integer partitions of odd numbers.

%I A300063 #12 May 22 2018 22:15:37
%S A300063 2,5,6,8,11,14,15,17,18,20,23,24,26,31,32,33,35,38,41,42,44,45,47,50,
%T A300063 51,54,56,58,59,60,65,67,68,69,72,73,74,77,78,80,83,86,92,93,95,96,97,
%U A300063 98,99,103,104,105,106,109,110,114,119,122,123,124,125,126,127
%N A300063 Heinz numbers of integer partitions of odd numbers.
%C A300063 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A300063 Alois P. Heinz, <a href="/A300063/b300063.txt">Table of n, a(n) for n = 1..10000</a>
%e A300063 15 is the Heinz number of (3,2), which has odd weight, so 15 belongs to the sequence.
%e A300063 Sequence of odd-weight partitions begins: (1) (3) (2,1) (1,1,1) (5) (4,1) (3,2) (7) (2,2,1) (3,1,1) (9) (2,1,1,1) (6,1).
%p A300063 a:= proc(n) option remember; local k; for k from 1+
%p A300063      `if`(n=1, 0, a(n-1)) while add(numtheory[pi]
%p A300063       (i[1])*i[2], i=ifactors(k)[2])::even do od; k
%p A300063     end:
%p A300063 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 22 2018
%t A300063 Select[Range[200],OddQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]
%Y A300063 Complement of A300061.
%Y A300063 Cf. A000041, A000720, A001222, A056239, A063834, A112798, A215366, A296150, A299757, A300056, A300060, A304662.
%K A300063 nonn
%O A300063 1,1
%A A300063 _Gus Wiseman_, Feb 23 2018