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A052002 Numbers with an odd number of partitions. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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).

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)

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 *)

PROG

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

(Haskell)

import Data.List (findIndices)

a052002 n = a052002_list !! (n-1)

a052002_list = findIndices odd a000041_list

-- Reinhard Zumkeller, Nov 03 2015

CROSSREFS

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

Sequence in context: A099562 A210450 A133896 * A247636 A070916 A078305

Adjacent sequences:  A051999 A052000 A052001 * A052003 A052004 A052005

KEYWORD

nonn,easy

AUTHOR

Patrick De Geest, Nov 15 1999

EXTENSIONS

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

STATUS

approved

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Last modified October 17 18:51 EDT 2019. Contains 328127 sequences. (Running on oeis4.)