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A300061 Heinz numbers of integer partitions of even numbers.

%I A300061 #14 May 22 2018 22:15:54
%S A300061 1,3,4,7,9,10,12,13,16,19,21,22,25,27,28,29,30,34,36,37,39,40,43,46,
%T A300061 48,49,52,53,55,57,61,62,63,64,66,70,71,75,76,79,81,82,84,85,87,88,89,
%U A300061 90,91,94,100,101,102,107,108,111,112,113,115,116,117,118,120
%N A300061 Heinz numbers of integer partitions of even numbers.
%C A300061 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A300061 Alois P. Heinz, <a href="/A300061/b300061.txt">Table of n, a(n) for n = 1..10000</a>
%e A300061 75 is the Heinz number of (3,3,2), which has even weight, so 75 belongs to the sequence.
%e A300061 Sequence of even-weight partitions begins: () (2) (1,1) (4) (2,2) (3,1) (2,1,1) (6) (1,1,1,1) (8) (4,2) (5,1) (3,3) (2,2,2) (4,1,1).
%p A300061 a:= proc(n) option remember; local k; for k from 1+
%p A300061      `if`(n=1, 0, a(n-1)) while add(numtheory[pi]
%p A300061       (i[1])*i[2], i=ifactors(k)[2])::odd do od; k
%p A300061     end:
%p A300061 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 22 2018
%t A300061 Select[Range[200],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]
%Y A300061 Complement of A300063.
%Y A300061 Cf. A000041, A000720, A001222, A056239, A063834, A100118, A112798, A122111, A215366, A296150, A299202, A299757, A300056, A300060, A304662.
%K A300061 nonn
%O A300061 1,2
%A A300061 _Gus Wiseman_, Feb 23 2018