A239241 -id:A239241 - cp's OEIS frontend

A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

0, 0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 43, 51, 61, 74, 91, 113, 144, 184, 239, 311, 407, 530, 692, 895, 1155, 1478, 1882, 2375, 2983, 3715, 4602, 5660, 6925, 8418, 10187, 12257, 14686, 17514, 20809, 24624, 29049, 34154, 40051, 46842, 54668, 63667

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even. This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239872.

			a(7) counts these 9 partitions of 15:  [12,1,2], [10,1,4], [10,3,2], [4,9,2], [8,1,6], [8,5,2], [8,3,4], [6,7,2], [6,5,4].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A239241, A239871, A239872, A239832.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(2*n+1$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n + 1]], {n, 0, 38}]  (* A239873 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2 Mod[i, 2] - 1)]]]]; a[n_] := b[2n+1, 2n+1, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

Original entry on oeis.org

1, 12, 70, 90, 112, 144, 286, 325, 462, 520, 525, 594, 646, 675, 832, 840, 1045, 1080, 1326, 1334, 1344, 1666, 1672, 1728, 1900, 2142, 2145, 2294, 2465, 2622, 2695, 2754, 3040, 3432, 3465, 3509, 3526, 3900, 3944, 4186, 4255, 4312, 4455, 4845, 4864, 4900, 4982

Offset: 1

Gus Wiseman, Oct 23 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    70: {1,3,4}
    90: {1,2,2,3}
   112: {1,1,1,1,4}
   144: {1,1,1,1,2,2}
   286: {1,5,6}
   325: {3,3,6}
   462: {1,2,4,5}
   520: {1,1,1,3,6}
   525: {2,3,3,4}
   594: {1,2,2,2,5}
   646: {1,7,8}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
   840: {1,1,1,2,3,4}
For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.

For prime factors instead of indices we have A019507.
Partitions of this type are counted by A239261.
For count instead of sum we have A325698, distinct A325700.
The LHS (sum of odd prime indices) is A366528, triangle A113685.
The RHS (sum of even prime indices) is A366531, triangle A113686.
These are the positions of zeros in A366749.
A000009 counts partitions into odd parts, ranked by A066208.
A035363 counts partitions into even parts, ranked by A066207.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A106529, A239241, A241638, A365067, A366533.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], Total[Select[prix[#],OddQ]]==Total[Select[prix[#],EvenQ]]&]

These are numbers k such that A346697(k) = A346698(k).

A255001 Number of partitions of 4n into distinct parts with equal sums of odd and even parts.

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 12, 15, 30, 40, 70, 96, 165, 216, 352, 486, 736, 988, 1518, 1998, 2944, 3952, 5607, 7488, 10614, 13916, 19305, 25536, 34854, 45568, 61864, 80240, 107640, 139776, 184832, 238680, 314628, 402800, 526176, 673652, 872592, 1110060, 1431704

Offset: 0

Alois P. Heinz, Feb 11 2015

nonn

			a(0) = 1: [], the empty partition.
a(1) = 0.
a(2) = 1: [4,3,1].
a(3) = 2: [6,5,1], [5,4,2,1].
a(4) = 4: [8,7,1], [8,5,3], [7,6,2,1], [6,5,3,2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000009, A000700, A069910, A239241, A249914.

g:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, g(n-i, i-1))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i-2))))
    end:
a:= n-> g(n$2)*b(2*n, 2*n-1):
seq(a(n), n=0..50);

g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, 1, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + If[i > n, 0, b[n - i, i - 2]]]];
a[n_] := g[n, n] b[2n, 2n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = A000009(n) * A069910(n) = A000009(n) * A000700(2n).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (16*sqrt(6)*n^(3/2)). - Vaclav Kotesovec, Dec 11 2020

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

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A239873 Number of strict partitions of 2n + 1 having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

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A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

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A255001 Number of partitions of 4n into distinct parts with equal sums of odd and even parts.

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A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

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