A280290 Numbers n such that number of partitions of n is even and number of partitions of n into distinct parts is even.

8, 9, 10, 11, 19, 21, 25, 27, 28, 30, 31, 34, 42, 45, 46, 47, 50, 55, 58, 59, 62, 64, 65, 66, 74, 75, 78, 79, 80, 84, 86, 94, 96, 97, 98, 101, 103, 106, 108, 109, 110, 112, 113, 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

Offset: 1

Author

Ilya Gutkovskiy, Dec 31 2016

Keywords

• nonn
• easy

Comments

Intersection of A001560 and A090864.

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

Examples

			8 is in the sequence because we have:
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number of partitions = 22 (is even)
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8 = 8
7 + 1 = 8
6 + 2 = 8
6 + 1 + 1 = 8
5 + 3 = 8
5 + 2 + 1 = 8
5 + 1 + 1 + 1 = 8
4 + 4 = 8
4 + 3 + 1 = 8
4 + 2 + 2 = 8
4 + 2 + 1 + 1 = 8
4 + 1 + 1 + 1 + 1 = 8
3 + 3 + 2 = 8
3 + 3 + 1 + 1 = 8
3 + 2 + 2 + 1 = 8
3 + 2 + 1 + 1 + 1 = 8
3 + 1 + 1 + 1 + 1 + 1 = 8
2 + 2 + 2 + 2 = 8
2 + 2 + 2 + 1 + 1 = 8
2 + 2 + 1 + 1 + 1 + 1 = 8
2 + 1 + 1 + 1 + 1 + 1 + 1 = 8
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
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number of partitions into distinct parts = 6 (is even)
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8 = 8
7 + 1 = 8
6 + 2 = 8
5 + 3 = 8
5 + 2 + 1 = 8
4 + 3 + 1 = 8
		

Links

• Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000009, A000035, A000041, A001560, A090864, A280288, A280289, A280291.

Programs

Mathematica

Select[Range[180], Mod[PartitionsP[#1], 2] == Mod[PartitionsQ[#1], 2] == 0 & ]