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.
Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - _N. J. A. Sloane_, Jun 03 2012
From _Gus Wiseman_, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
(11) (22) (33) (44)
(211) (321) (422)
(1111) (2211) (431)
(3111) (2222)
(21111) (3221)
(111111) (3311)
(4211)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
(End)
g:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
end:
b:= proc(n, i, s) option remember;
`if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,
b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,
max(x, n-i-x)), `if`(xn, NULL, max(x-i, n-x))}[], s)))))
end:
a:= n-> b(2*n, n, {n}):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 10 2012
b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *)
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s];
Table[Length[Select[IntegerPartitions[2n],MemberQ[Total/@subptns[#],n]&]],{n,10}] (* Gus Wiseman, Oct 27 2022 *)
from itertools import combinations_with_replacement from sympy.utilities.iterables import partitions def A002219(n): return len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023
The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
0 1
1 1
2 2 1
5 3 3
8 4 9 1
17 6 16 1 2
24 7 33 4 9
46 11 52 3 18 1 4
64 12 91 6 38 3 15 1 1
107 17 138 9 68 2 28 2 12 0 2
147 19 219 12 117 6 56 3 34 2 9 0 3
Row n = 4 counts the following partitions:
(8) (44) (431) (221111)
(71) (3311) (422)
(62) (2222) (4211)
(611) (11111111) (41111)
(53) (3221)
(521) (32111)
(5111) (311111)
(332) (22211)
(2111111)
t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]
The a(3600) = 5 divisors, their prime indices, and the prime indices of their quotients:
45: {2,2,3} * {1,1,1,1,3}
50: {1,3,3} * {1,1,1,2,2}
60: {1,1,2,3} * {1,1,2,3}
72: {1,1,1,2,2} * {1,3,3}
80: {1,1,1,1,3} * {2,2,3}
sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Table[Length[Select[Divisors[n],sumprix[#]==sumprix[n]/2&]],{n,100}]
A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); A357879(n) = sumdiv(n,d, A056239(d)==A056239(n/d)); \\ Antti Karttunen, Jan 20 2025
The array begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ... 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, ... 1, 11, 53, 173, 448, 994, 1974, 3606, 6171, 10021, ... 1, 15, 89, 343, 1022, 2562, 5670, 11418, 21351, 37609, ... 1, 22, 167, 778, 2710, 7764, 19314, 43164, 88671, 170170, ... ...
with(combinat):
g:= proc(n, m) option remember;
`if`(m>1, map(x-> map(y-> sort([x[], y[]]), g(n, 1))[],
g(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
{partition(n)[]}), {[]}))
end:
T:= (n, m)-> nops(g(n, m)):
seq(seq(T(d-m, m), m=0..d-1), d=1..12); # Alois P. Heinz, Jul 11 2012
T[n_, m_] := Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ ip]; i[0]=1; Table[Join[Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k-1], lg}, {k, 1, m}]]] // Flatten[#, m-1]& // Union // Length]; T[_, 0] = 1;
Table[T[n-m, m], {n, 1, 12}, {m, 0, n - 1}] // Flatten (* Jean-François Alcover, May 25 2016 *)
From _Gus Wiseman_, Apr 20 2024: (Start)
The a(1) = 1 through a(3) = 10 triquanimous partitions:
(111) (222) (333)
(2211) (3321)
(21111) (32211)
(111111) (33111)
(222111)
(321111)
(2211111)
(3111111)
(21111111)
(111111111)
(End)
Row T(5) = {0, 2, 9, 10, 3}, so P_5(k) = (1/4!)(2k + 9k^2 + 10k^3 + 3k^4), which gives 1, 7, 25, 65, 140, 266, ..., that is A001296 (row 5 of A213086), for k >=1.
Triangle begins:
{1},
{0, 1},
{0, 1, 1},
{0, 1, 3, 2},
{0, 2, 9, 10, 3},
{0, 2, 25, 50, 35, 8},
{0, -12, 86, 270, 260, 102, 14},
...
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