cp's OEIS Frontend

A280291 Numbers n such that number of partitions of n is odd and number of partitions of n into distinct parts is odd.

%I A280291 #11 Feb 16 2025 08:33:38
%S A280291 0,1,5,7,12,35,51,77,92,145,155,210,222,287,301,330,345,376,392,425,
%T A280291 442,477,495,610,672,737,805,852,876,1190,1247,1335,1426,1617,1717,
%U A280291 1855,1962,2035,2147,2542,2625,2795,2882,3197,3337,3480,3626,3775,3876,4030,4347,4510,4565,4845,4902
%N A280291 Numbers n such that number of partitions of n is odd and number of partitions of n into distinct parts is odd.
%C A280291 Intersection of A001318 and A052002.
%C A280291 Numbers n such that A000035(A000041(n)) = 1 and A000035(A000009(n)) = 1.
%H A280291 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 A280291 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%e A280291 7 is in the sequence because we have:
%e A280291 ----------------------------------
%e A280291 number of partitions = 15 (is odd)
%e A280291 ----------------------------------
%e A280291 7 = 7
%e A280291 6 + 1 = 7
%e A280291 5 + 2 = 7
%e A280291 5 + 1 + 1 = 7
%e A280291 4 + 3 = 7
%e A280291 4 + 2 + 1 = 7
%e A280291 4 + 1 + 1 + 1 = 7
%e A280291 3 + 3 + 1 = 7
%e A280291 3 + 2 + 2 = 7
%e A280291 3 + 2 + 1 + 1 = 7
%e A280291 3 + 1 + 1 + 1 + 1 = 7
%e A280291 2 + 2 + 2 + 1 = 7
%e A280291 2 + 2 + 1 + 1 + 1 = 7
%e A280291 2 + 1 + 1 + 1 + 1 + 1 = 7
%e A280291 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
%e A280291 -----------------------------------------------------
%e A280291 number of partitions into distinct parts = 5 (is odd)
%e A280291 -----------------------------------------------------
%e A280291 7 = 7
%e A280291 6 + 1 = 7
%e A280291 5 + 2 = 7
%e A280291 4 + 3 = 7
%e A280291 4 + 2 + 1 = 7
%t A280291 Join[{0}, Select[Range[5000], Mod[PartitionsP[#1], 2] == Mod[PartitionsQ[#1], 2] == 1 & ]]
%Y A280291 Cf. A000009, A000035, A000041, A001318, A052002, A280288, A280289, A280290.
%K A280291 nonn,easy
%O A280291 1,3
%A A280291 _Ilya Gutkovskiy_, Dec 31 2016
%E A280291 a(1)=0 inserted by _Alois P. Heinz_, Dec 31 2016