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 A052002 Numbers with an odd number of partitions. 13
 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) EXAMPLE 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) MAPLE N:= 1000: # to get all terms <= N V:= Vector(N+1): V:= 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, 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. 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. 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 September 30 05:03 EDT 2020. Contains 337435 sequences. (Running on oeis4.)