A052002 Numbers with an odd number of partitions.

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, 43, 44, 48, 49, 51, 52, 53, 54, 56, 60, 61, 63, 67, 68, 69, 71, 72, 73, 76, 77, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 95, 99, 102, 104, 105, 107, 111, 114, 115, 118, 119, 121

Offset: 1

Author

Patrick De Geest, Nov 15 1999

Keywords

• nonn
• easy

Comments

A052003(n) = A000041(a(n+1)). - Reinhard Zumkeller, Nov 03 2015

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

Examples

			From _Gus Wiseman_, Jan 13 2020: (Start)
The partitions of the initial terms are:
  (1)  (3)    (4)     (5)      (6)       (7)
       (21)   (22)    (32)     (33)      (43)
       (111)  (31)    (41)     (42)      (52)
              (211)   (221)    (51)      (61)
              (1111)  (311)    (222)     (322)
                      (2111)   (321)     (331)
                      (11111)  (411)     (421)
                               (2211)    (511)
                               (3111)    (2221)
                               (21111)   (3211)
                               (111111)  (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)
(End)
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).

Crossrefs

Cf. A000041, A000700, A001560, A052001, A052003.

The strict version is A001318, with complement A090864.

The version for prime instead of odd numbers is A046063.

The version for squarefree instead of odd numbers is A038630.

The version for set partitions appears to be A032766.

The version for factorizations is A331050.

The version for strict factorizations is A331230.

Programs

Haskell

import Data.List (findIndices)
a052002 n = a052002_list !! (n-1)
a052002_list = findIndices odd a000041_list
-- Reinhard Zumkeller, Nov 03 2015

Maple

N:= 1000: # to get all terms <= N
V:= Vector(N+1):
V[1]:= 1:
for i from 1 to (N+1)/2  do
  V[2*i..N+1]:= V[2*i..N+1] + V[1..N-2*i+2] mod 2
od:
select(t -> V[t+1]=1, [$1..N]); # Robert Israel, Jan 22 2017

Mathematica

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[2, k], {k, 0, 1}] (* Clark Kimberling, Jan 05 2014 *)

PARI

for(n=0, 200, if(numbpart(n)%2==1, print1(n", "))) \\ Altug Alkan, Nov 02 2015

Formula

From Peter Bala, Jan 22 2016: (Start)

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)

Extensions

Offset corrected and b-file adjusted by Reinhard Zumkeller, Nov 03 2015