A154798 -id:A154798 - cp's OEIS frontend

A163288 a(n) = A154798(n)/2.

1, 11, 21, 501, 1218, 1859, 2802, 6155, 18669, 26587, 52779, 102113, 357610, 650078, 870815, 1161760, 2043984, 3544750, 6066082, 7898238, 13271830, 17131481, 46334860, 59057152, 75099068, 95284646, 192138168, 241751422, 303581873, 380501078, 594454124, 922174780, 1145660456

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000041, A154798.

(1/2)*Select[Table[PartitionsP[n], {n, 0, 500, 2}], EvenQ] (* G. C. Greubel, Dec 17 2016 *)

select(x->!(x%2), vector(80, n, numbpart(2*n)))/2 \\ G. C. Greubel, Dec 17 2016

Terms a(13) onward from G. C. Greubel, Dec 17 2016

A154795 Odd partition numbers of odd numbers.

Original entry on oeis.org

1, 3, 7, 15, 101, 297, 1255, 4565, 10143, 14883, 21637, 31185, 44583, 63261, 173525, 239943, 329931, 1121505, 1505499, 2679689, 3554345, 4697205, 6185689, 10619863, 18004327, 23338469, 30167357, 38887673, 49995925, 64112359, 82010177

Offset: 1

Omar E. Pol, Jan 26 2009

nonn

Odd numbers in A058695.

			7 is in the sequence because the odd number 5 has partition number 7 (5,41,32,311,2221,22111,1111111). - _Emeric Deutsch_, Aug 02 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A005408, A058695, A154796, A154797, A154798.

aa:= proc(n, i) if n=0 then 1 elif n<0 or i=0 then 0 else aa(n,i):= aa(n, i-1) +aa(n-i, i) fi end: a:= proc(n) local k; if n>1 then a(n-1) fi; for k from `if`(n=1, 1, b(n-1)+2) by 2 while irem(aa(k, k), 2)=0 do od; b(n):= k; aa(k, k) end: seq(a(n), n=1..40); # Alois P. Heinz, Jul 28 2009
with(combinat): a := proc (n) if `mod`(numbpart(2*n-1), 2) = 1 then numbpart(2*n-1) else end if end proc: seq(a(n), n = 1 .. 50); # Emeric Deutsch, Aug 02 2009

Reap[Do[If[OddQ[p = PartitionsP[n]], Sow[p]], {n, 1, 99, 2}]][[2, 1]] (* Jean-François Alcover, Aug 31 2015 *)

More terms from Alois P. Heinz, Jul 28 2009

A127219 Even numbers with an even number of partitions.

Original entry on oeis.org

2, 8, 10, 22, 26, 28, 30, 34, 40, 42, 46, 50, 58, 62, 64, 66, 70, 74, 78, 80, 84, 86, 94, 96, 98, 100, 106, 108, 110, 112, 116, 120, 122, 124, 126, 128, 130, 136, 142, 154, 158, 160, 170, 174, 176, 180, 184, 198, 200, 206, 224, 228, 230, 236, 246, 248

Offset: 1

Zak Seidov, Mar 28 2007

nonn

			10 is in the sequence because the number of partitions of 10 is equal to 42 and both 10 and 42 are even numbers. - _Omar E. Pol_, Mar 18 2012

Cf. A000041, A005843, A154798.

with(combinat): a:=proc(n) if numbpart(2*n) mod 2 = 0 then 2*n else fi end: seq(a(n),n=1..100); # Emeric Deutsch, Mar 31 2007

Select[Range[2, 250, 2], EvenQ[PartitionsP[#]]&] (* Jean-François Alcover, Aug 28 2024 *)

More terms a(50)-a(56) from Omar E. Pol, Mar 18 2012

A154796 Even partition numbers of odd numbers.

Original entry on oeis.org

30, 56, 176, 490, 792, 1958, 3010, 6842, 89134, 124754, 451276, 614154, 831820, 2012558, 8118264, 13848650, 133230930, 214481126, 271248950, 541946240, 851376628, 1327710076, 3163127352, 4835271870, 5964539504, 7346629512

Offset: 1

Omar E. Pol, Jan 26 2009

nonn

Even numbers in A058695.

			The even number 30 is in the sequence as the partition number of the odd number 9. - _Emeric Deutsch_, Aug 02 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A005408, A058695, A154795, A154797, A154798.

aa:= proc(n, i) if n=0 then 1 elif n<0 or i=0 then 0 else aa(n,i):= aa(n, i-1) +aa(n-i, i) fi end: a:= proc(n) local k; if n>1 then a(n-1) fi; for k from `if`(n=1, 1, b(n-1)+2) by 2 while irem(aa(k, k), 2)=1 do od; b(n):= k; aa(k, k) end: seq(a(n), n=1..40); # Alois P. Heinz, Jul 28 2009
with(combinat): a := proc (n) if `mod`(numbpart(2*n-1), 2) = 0 then numbpart(2*n-1) else end if end proc: seq(a(n), n = 1 .. 70); # Emeric Deutsch, Aug 02 2009

Reap[Do[If[EvenQ[p = PartitionsP[n]], Sow[p]], {n, 1, 199, 2}]][[2, 1]] (* Jean-François Alcover, Nov 11 2015 *)
Select[PartitionsP[Range[1,201,2]],EvenQ] (* Harvey P. Dale, Apr 03 2019 *)

lista(nn) = for (n=1, nn, if (((p = numbpart(2*n+1)) % 2) == 0, print1(p, ", "))); \\ Michel Marcus, Dec 19 2016

More terms from Alois P. Heinz, Jul 28 2009

A154797 Odd partition numbers of even numbers.

Original entry on oeis.org

1, 5, 11, 77, 135, 231, 385, 627, 1575, 8349, 17977, 26015, 75175, 147273, 281589, 386155, 526823, 966467, 3087735, 5392783, 9289091, 20506255, 44108109, 56634173, 72533807, 241265379, 304801365, 952050665, 1482074143, 6620830889

Offset: 1

Omar E. Pol, Jan 26 2009

nonn

Odd numbers in A058696.

			The odd number 5 is in the sequence as the partition number of the even number 4: (4, 3+1, 2+2, 2+1+1, 1+1+1+1). - _Emeric Deutsch_, Aug 02 2009

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A005843, A058696, A154795, A154796, A154798.

aa:= proc(n, i) if n=0 then 1 elif n<0 or i=0 then 0 else aa(n,i):= aa(n, i-1) +aa(n-i, i) fi end: a:= proc(n) local k; if n>1 then a(n-1) fi; for k from `if`(n=1, 0, b(n-1)+2) by 2 while irem(aa(k, k), 2)=0 do od; b(n):= k; aa(k, k) end: seq(a(n), n=1..40); # Alois P. Heinz, Jul 28 2009
with(combinat): a := proc (n) if `mod`(numbpart(2*n), 2) = 1 then numbpart(2*n) else end if end proc: seq(a(n), n = 0 .. 70); # Emeric Deutsch, Aug 02 2009

Select[PartitionsP[2*Range[0,100]],OddQ] (* Harvey P. Dale, Nov 30 2014 *)

More terms from Alois P. Heinz, Jul 28 2009

A194798 Numbers n having the same parity as the number of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 13, 17, 22, 23, 26, 28, 29, 30, 33, 34, 35, 37, 39, 40, 41, 42, 43, 46, 49, 50, 51, 53, 58, 61, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 77, 78, 80, 81, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 96, 98, 99, 100, 105, 106, 107, 108, 110, 111

Offset: 1

Omar E. Pol, Jan 29 2012

Odd positive integers with an odd number of partitions and even positive integers with an even number of partitions. - Omar E. Pol, Mar 17 2012

Union of A067567 and A127219. Note that the union of A163096 and A163097 gives A209920 and the union of A209920 and this sequence gives A001477. - Omar E. Pol, Mar 22 2012

			10 is in the sequence because the number of partitions of 10 is equal to 42 and both 10 and 42 have the same parity.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659, A209920.

with(combinat):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while irem(k+numbpart(k), 2)=1 do od; k
    end:
seq(a(n), n=1..80); # Alois P. Heinz, Mar 16 2012

Select[Range[200], Mod[PartitionsP[#] - #, 2] == 0 &] (* T. D. Noe, Mar 16 2012 *)

More terms from Alois P. Heinz, Mar 16 2012

A209920 Numbers n having distinct parity as the number of partitions of n.

Original entry on oeis.org

0, 4, 6, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 24, 25, 27, 31, 32, 36, 38, 44, 45, 47, 48, 52, 54, 55, 56, 57, 59, 60, 65, 68, 72, 75, 76, 79, 82, 88, 90, 92, 97, 101, 102, 103, 104, 109, 113, 114, 117, 118, 125, 129, 131, 132, 133, 134, 135, 137, 138, 140

Offset: 1

Omar E. Pol, Mar 16 2012

nonn

Odd positive integers with an even number of partitions and nonnegative even integers with an odd number of partitions. Union of A163097 and A163096. Note that the union of A067567 and A127219 gives A194798 and the union of A194798 and this sequence gives A001477.

			4 is in the sequence because the number of partitions of 4 is equal to 5 and the parity of 4 is distinct to the parity of 5 because 4 is even and 5 is odd.
9 is in the sequence because the number of partitions of 9 is equal to 30 and the parity of 9 is distinct to the parity of 30 because 9 is odd and 30 is even.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Complement of A194798.

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659.

A209658 Partition numbers p(n) having the same parity as n.

Original entry on oeis.org

1, 2, 3, 7, 15, 22, 42, 101, 297, 1002, 1255, 2436, 3718, 4565, 5604, 10143, 12310, 14883, 21637, 31185, 37338, 44583, 53174, 63261, 105558, 173525, 204226, 239943, 329931, 715220, 1121505, 1300156, 1505499, 1741630, 2323520, 2679689, 3554345

Offset: 1

Omar E. Pol, Mar 22 2012

nonn

Union of A154795 and A154798. The union of A209659 and this sequence gives A000041.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A194798, A209659, A209920.

A209659 Partition numbers p(n) having opposite parity of n.

Original entry on oeis.org

1, 5, 11, 30, 56, 77, 135, 176, 231, 385, 490, 627, 792, 1575, 1958, 3010, 6842, 8349, 17977, 26015, 75175, 89134, 124754, 147273, 281589, 386155, 451276, 526823, 614154, 831820, 966467, 2012558, 3087735, 5392783, 8118264, 9289091, 13848650

Offset: 1

Omar E. Pol, Mar 22 2012

nonn

Union of A154797 and A154796. The union of this sequence and A209658 gives A000041.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A194798, A209658, A209920.

A163098 a(n) = A127219(n)/2.

Original entry on oeis.org

1, 4, 5, 11, 13, 14, 15, 17, 20, 21, 23, 25, 29, 31, 32, 33, 35, 37, 39, 40, 42, 43, 47, 48, 49, 50, 53, 54, 55, 56, 58, 60, 61, 62, 63, 64, 65, 68, 71, 77, 79, 80, 85, 87, 88, 90, 92, 99, 100, 103, 112, 114, 115, 118, 123, 124, 128, 129, 130, 131, 132, 134, 137, 139, 140

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Cf. A000041, A127219, A154798, A163288.

Contribution from R. J. Mathar, Oct 09 2010: (Start)
A127219 := proc(n) option remember; if n = 1 then 2; else for a from procname(n-1)+2 by 2 do if type(combinat[numbpart](a) ,'even') then return a; fi; end do; fi ; end proc:
A163098 := proc(n) A127219(n)/2 ; end proc;
seq(A163098(n),n=1..100) ; (End)

More terms from R. J. Mathar, Oct 09 2010

cp's OEIS Frontend

A163288 a(n) = A154798(n)/2.

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A154795 Odd partition numbers of odd numbers.

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A127219 Even numbers with an even number of partitions.

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A154796 Even partition numbers of odd numbers.

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Maple

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A154797 Odd partition numbers of even numbers.

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Maple

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A194798 Numbers n having the same parity as the number of partitions of n.

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Maple

Mathematica

Extensions

A209920 Numbers n having distinct parity as the number of partitions of n.

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A209658 Partition numbers p(n) having the same parity as n.

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