%I #5 Dec 31 2016 06:41:09
%S 8,9,10,11,19,21,25,27,28,30,31,34,42,45,46,47,50,55,58,59,62,64,65,
%T 66,74,75,78,79,80,84,86,94,96,97,98,101,103,106,108,109,110,112,113,
%U 116,120,122,124,125,128,129,130,131,133,135,136,137,141,142,147,149,151,153,154,158,160,163,167,170,171,174,175,179,180
%N Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is even.
%C Intersection of A001560 and A090864.
%C Numbers n such that A000035(A000041(n)) = 0 and A000035(A000009(n)) = 0.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%e 8 is in the sequence because we have:
%e -----------------------------------
%e number of partitions = 22 (is even)
%e -----------------------------------
%e 8 = 8
%e 7 + 1 = 8
%e 6 + 2 = 8
%e 6 + 1 + 1 = 8
%e 5 + 3 = 8
%e 5 + 2 + 1 = 8
%e 5 + 1 + 1 + 1 = 8
%e 4 + 4 = 8
%e 4 + 3 + 1 = 8
%e 4 + 2 + 2 = 8
%e 4 + 2 + 1 + 1 = 8
%e 4 + 1 + 1 + 1 + 1 = 8
%e 3 + 3 + 2 = 8
%e 3 + 3 + 1 + 1 = 8
%e 3 + 2 + 2 + 1 = 8
%e 3 + 2 + 1 + 1 + 1 = 8
%e 3 + 1 + 1 + 1 + 1 + 1 = 8
%e 2 + 2 + 2 + 2 = 8
%e 2 + 2 + 2 + 1 + 1 = 8
%e 2 + 2 + 1 + 1 + 1 + 1 = 8
%e 2 + 1 + 1 + 1 + 1 + 1 + 1 = 8
%e 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
%e -------------------------------------------------------
%e number of partitions into distinct parts = 6 (is even)
%e -------------------------------------------------------
%e 8 = 8
%e 7 + 1 = 8
%e 6 + 2 = 8
%e 5 + 3 = 8
%e 5 + 2 + 1 = 8
%e 4 + 3 + 1 = 8
%t Select[Range[180], Mod[PartitionsP[#1], 2] == Mod[PartitionsQ[#1], 2] == 0 & ]
%Y Cf. A000009, A000035, A000041, A001560, A090864, A280288, A280289, A280291.
%K nonn,easy
%O 1,1
%A _Ilya Gutkovskiy_, Dec 31 2016