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A280289 Numbers n such that number of partitions of n is odd and number of partitions of n into distinct parts is even.

%I A280289 #6 Feb 16 2025 08:33:38
%S A280289 3,4,6,13,14,16,17,18,20,23,24,29,32,33,36,37,38,39,41,43,44,48,49,52,
%T A280289 53,54,56,60,61,63,67,68,69,71,72,73,76,81,82,83,85,87,88,89,90,91,93,
%U A280289 95,99,102,104,105,107,111,114,115,118,119,121,123,127,132,134,138,139,140,143,144,146,148,150,152,156,157,159
%N A280289 Numbers n such that number of partitions of n is odd and number of partitions of n into distinct parts is even.
%C A280289 Intersection of A052002 and A090864.
%C A280289 Numbers n such that A000035(A000041(n)) = 1 and A000035(A000009(n)) = 0.
%H A280289 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>, <a href="https://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>
%H A280289 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%e A280289 6 is in the sequence because we have:
%e A280289 ----------------------------------
%e A280289 number of partitions = 11 (is odd)
%e A280289 ----------------------------------
%e A280289 6 = 6
%e A280289 5 + 1 = 6
%e A280289 4 + 2 = 6
%e A280289 4 + 1 + 1 = 6
%e A280289 3 + 3 = 6
%e A280289 3 + 2 + 1 = 6
%e A280289 3 + 1 + 1 + 1 = 6
%e A280289 2 + 2 + 2 = 6
%e A280289 2 + 2 + 1 + 1 = 6
%e A280289 2 + 1 + 1 + 1 + 1 = 6
%e A280289 1 + 1 + 1 + 1 + 1 + 1 = 6
%e A280289 ------------------------------------------------------
%e A280289 number of partitions into distinct parts = 4 (is even)
%e A280289 ------------------------------------------------------
%e A280289 6 = 6
%e A280289 5 + 1 = 6
%e A280289 4 + 2 = 6
%e A280289 3 + 2 + 1 = 6
%t A280289 Select[Range[160], Mod[PartitionsP[#1], 2] == 1 && Mod[PartitionsQ[#1], 2] == 0 & ]
%Y A280289 Cf. A000009, A000035, A000041, A052002, A090864, A280288, A280290, A280291.
%K A280289 nonn,easy
%O A280289 1,1
%A A280289 _Ilya Gutkovskiy_, Dec 31 2016