This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A052002 #50 Jan 13 2025 10:49:52
%S A052002 0,1,3,4,5,6,7,12,13,14,16,17,18,20,23,24,29,32,33,35,36,37,38,39,41,
%T A052002 43,44,48,49,51,52,53,54,56,60,61,63,67,68,69,71,72,73,76,77,81,82,83,
%U A052002 85,87,88,89,90,91,92,93,95,99,102,104,105,107,111,114,115,118,119,121
%N A052002 Numbers with an odd number of partitions.
%C A052002 A052003(n) = A000041(a(n+1)). - _Reinhard Zumkeller_, Nov 03 2015
%C A052002 Also, numbers having an odd number of partitions into distinct odd parts; that is, numbers m such that A000700(m) is odd. For example, 16 is in the list since 16 has 5 partitions into distinct odd parts, namely, 1 + 15, 3 + 13, 5 + 11, 7 + 9 and 1 + 3 + 5 + 7. See Formula section for a proof. - _Peter Bala_, Jan 22 2017
%H A052002 Clark Kimberling, <a href="/A052002/b052002.txt">Table of n, a(n) for n = 1..1000</a>
%H A052002 O. Kolberg, <a href="http://www.mscand.dk/article/view/10584/8605">Note on the parity of the partition function</a>, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).
%F A052002 From _Peter Bala_, Jan 22 2016: (Start)
%F A052002 Sum_{n>=0} x^a(n) = (1 + x)*(1 + x^3)*(1 + x^5)*... taken modulo 2. Proof: Product_{n>=1} 1 + x^(2*n-1) = Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(2*n-1)) = Product_{n>=1} (1 - x^(2*n))*(1 - x^(4*n-2))/( (1 - x^(2*n)) * (1 - x^(2*n-1)) ) = ( 1 + 2*Sum_{n>=1} (-1)^n*x^(2*n^2) )/(Product_{n>=1} (1 - x^n)) == 1/( Product_{n>=1} (1 - x^n) ) (mod 2). (End)
%e A052002 From _Gus Wiseman_, Jan 13 2020: (Start)
%e A052002 The partitions of the initial terms are:
%e A052002 (1) (3) (4) (5) (6) (7)
%e A052002 (21) (22) (32) (33) (43)
%e A052002 (111) (31) (41) (42) (52)
%e A052002 (211) (221) (51) (61)
%e A052002 (1111) (311) (222) (322)
%e A052002 (2111) (321) (331)
%e A052002 (11111) (411) (421)
%e A052002 (2211) (511)
%e A052002 (3111) (2221)
%e A052002 (21111) (3211)
%e A052002 (111111) (4111)
%e A052002 (22111)
%e A052002 (31111)
%e A052002 (211111)
%e A052002 (1111111)
%e A052002 (End)
%p A052002 N:= 1000: # to get all terms <= N
%p A052002 V:= Vector(N+1):
%p A052002 V[1]:= 1:
%p A052002 for i from 1 to (N+1)/2 do
%p A052002 V[2*i..N+1]:= V[2*i..N+1] + V[1..N-2*i+2] mod 2
%p A052002 od:
%p A052002 select(t -> V[t+1]=1, [$1..N]); # _Robert Israel_, Jan 22 2017
%t A052002 f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
%t A052002 Table[f[2, k], {k, 0, 1}] (* _Clark Kimberling_, Jan 05 2014 *)
%o A052002 (PARI) for(n=0, 200, if(numbpart(n)%2==1, print1(n", "))) \\ _Altug Alkan_, Nov 02 2015
%o A052002 (Haskell)
%o A052002 import Data.List (findIndices)
%o A052002 a052002 n = a052002_list !! (n-1)
%o A052002 a052002_list = findIndices odd a000041_list
%o A052002 -- _Reinhard Zumkeller_, Nov 03 2015
%Y A052002 Cf. A000041, A000700, A001560, A052001, A052003.
%Y A052002 The strict version is A001318, with complement A090864.
%Y A052002 The version for prime instead of odd numbers is A046063.
%Y A052002 The version for squarefree instead of odd numbers is A038630.
%Y A052002 The version for set partitions appears to be A032766.
%Y A052002 The version for factorizations is A331050.
%Y A052002 The version for strict factorizations is A331230.
%K A052002 nonn,easy
%O A052002 1,3
%A A052002 _Patrick De Geest_, Nov 15 1999
%E A052002 Offset corrected and b-file adjusted by _Reinhard Zumkeller_, Nov 03 2015