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Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is odd.
3

%I #5 Dec 31 2016 06:40:53

%S 2,15,22,26,40,57,70,100,117,126,176,187,247,260,532,551,590,651,715,

%T 782,925,950,1001,1027,1080,1107,1162,1276,1365,1457,1520,1552,1650,

%U 1751,1820,1926,2072,2185,2262,2301,2380,2420,2501,2667,2752,2926,3015,3060,3151,3290,3432,3577,3725,3927,4082,4187,4240,4401

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

%C Intersection of A001318 and A001560.

%C Numbers n such that A000035(A000041(n)) = 0 and A000035(A000009(n)) = 1.

%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 15 is in the sequence because we have:

%e ------------------------------------

%e number of partitions = 176 (is even)

%e ------------------------------------

%e 15 = 15

%e 14 + 1 = 15

%e 13 + 2 = 15

%e 13 + 1 + 1 = 15

%e 12 + 3 = 15

%e 12 + 2 + 1 = 15

%e 12 + 1 + 1 + 1 = 15

%e 11 + 4 = 15

%e 11 + 3 + 1 = 15

%e 11 + 2 + 2 = 15

%e 11 + 2 + 1 + 1 = 15

%e 11 + 1 + 1 + 1 + 1 = 15

%e ...

%e ------------------------------------------------------

%e number of partitions into distinct parts = 27 (is odd)

%e ------------------------------------------------------

%e 15 = 15

%e 14 + 1 = 15

%e 13 + 2 = 15

%e 12 + 3 = 15

%e 12 + 2 + 1 = 15

%e 11 + 4 = 15

%e 11 + 3 + 1 = 15

%e 10 + 5 = 15

%e 10 + 4 + 1 = 15

%e 10 + 3 + 2 = 15

%e ...

%t Select[Range[4500], Mod[PartitionsP[#1], 2] == 0 && Mod[PartitionsQ[#1], 2] == 1 & ]

%Y Cf. A000009, A000035, A000041, A001318, A001560, A280289, A280290, A280291.

%K nonn,easy

%O 1,1

%A _Ilya Gutkovskiy_, Dec 31 2016