cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 70, 0, 0, 0, 165, 0, 0, 0, 330, 0, 0, 0, 704, 0, 0, 0, 1380, 0, 0, 0, 2688, 0, 0, 0, 4984, 0, 0, 0, 9394, 0, 0, 0, 16665, 0, 0, 0, 29970, 0, 0, 0, 52096, 0, 0, 0, 90090, 0, 0, 0, 152064, 0, 0, 0
Offset: 0

Views

Author

Clark Kimberling, Mar 13 2014

Keywords

Examples

			a(8) counts these 4 partitions:  431, 41111, 3221, 221111.
From _Gus Wiseman_, Oct 24 2023: (Start)
The a(0) = 1 through a(12) = 12 partitions:
  ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)
                               (3221)             (651)
                               (41111)            (4332)
                               (221111)           (5421)
                                                  (33222)
                                                  (52221)
                                                  (63111)
                                                  (432111)
                                                  (3222111)
                                                  (6111111)
                                                  (42111111)
                                                  (222111111)
(End)

Links

Crossrefs

Cf. A239259, A239260, A239262, A239263, A000041.
The LHS (sum of odd parts) is counted by A113685.
The RHS (sum of even parts) is counted by A113686.
Without all the zeros we have a(4n) = A249914(n).
The strict case (without zeros) is A255001.
The Heinz numbers of these partitions are A366748, see also A019507.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.

Programs

  • Mathematica
    z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
(* Peter J. C. Moses, Mar 12 2014 *)

Formula

A239260(n) + a(n) + A239262(n) = A000041(n).
From David A. Corneth, Oct 25 2023: (Start)
a(4*n) = A000009(2*n) * A000041(n) for n >= 0.
a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)

Extensions

More terms from Alois P. Heinz, Mar 15 2014

A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

Original entry on oeis.org

72, 240, 672, 800, 2240, 4224, 5184, 6272, 9984, 14080, 17280, 33280, 39424, 48384, 52224, 57600, 93184, 116736, 161280, 174080, 192000, 247808, 304128, 373248, 389120, 451584, 487424, 537600, 565248, 585728, 640000, 718848, 1013760, 1089536, 1244160, 1384448
Offset: 1

Views

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Keywords

Examples

			6272 = 2*2*2*2*2*2*2*7*7 is droll since 2+2+2+2+2+2+2 = 14 = 7+7.

Links

Crossrefs

For count instead of sum we have A072978.
Partitions of this type are counted by A239261, without zero terms A249914.
For prime indices instead of factors we have A366748, zeros of A366749.
The LHS is A366839 with alternating zeros, for indices A366531, triangle A113686.
The RHS is A366840, for indices A366528, triangle A113685.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
A112798 lists prime indices, 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, A100367, A106529, A195017, A239241, A241638, A325698, A325700, A366533.

Programs

  • Maple
    f:= proc(k, m) # numbers whose sum of prime factors >= m is k; m is prime
   local S,p,j;
   option remember;
   if k = 0 then return [1]
   elif m > k then return []
   fi;
   S:= NULL:
   p:= nextprime(m);
   for j from k by -m to 0 do
     S:= S, op(map(`*`,  procname(j,p) , m^((k-j)/m)))
   od;
   [S]
end proc:
g:= proc(N) local m,R;
  R:= NULL;
  for m from 1 while 2^m < N do
   R:= R, op(map(`*`,select(`<=`,f(2*m,3), N/2^m),2^m));
  od;
  sort([R])
end proc:
g(10^8); # Robert Israel, Feb 20 2025
  • Mathematica
    Select[Range[2, 2*10^6, 2], First[#] == Total[Rest[#]] & [Times @@@ FactorInteger[#]] &] (* Paolo Xausa, Feb 19 2025 *)
  • PARI
    isok(n) = {if (n % 2, return (0)); f = factor(n); return (2*f[1,2] == sum(i=2, #f~, f[i,1]*f[i,2]));} \\ Michel Marcus, Jun 21 2013

Formula

These are even numbers k such that A366839(k/2) = A366840(k). - Gus Wiseman, Oct 25 2023 (corrected Feb 19 2025)

Extensions

Name edited by Paolo Xausa, Feb 19 2025

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

Views

Author

Alois P. Heinz, Feb 11 2015

Keywords

Examples

			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].

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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
Showing 1-3 of 3 results.

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