A331230 -id:A331230 - cp's OEIS frontend

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

Patrick De Geest, Nov 15 1999

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

			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)

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).

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.

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

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

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

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

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)

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

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[strfacs[#]]]&]

A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319238 in having 300.

The version for integer partitions is A001560.
The version for strict integer partitions is A090864.
The version for set partitions appears to be A016789.
The non-strict version is A331051.
The version for primes (instead of evens) is A331201.
The odd version is A331230.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A045783, A318286, A328966, A330973, A330991, A330997, A331022, A331023/A331024, A331050, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],EvenQ[Length[strfacs[#]]]&]

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A052002 Numbers with an odd number of partitions.

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A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

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A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

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