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A052002
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Numbers with an odd number of partitions.
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15
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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EXAMPLE
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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)
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MAPLE
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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:
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MATHEMATICA
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f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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