A235130 -id:A235130 - cp's OEIS frontend

A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304

Offset: 1

N. J. A. Sloane

			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)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.

Column m=2 of A213086.
Bisection of A276107.
Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.
The strict version is A237258, ranked by A357854.
Ranked by A357976 = positions of nonzero terms in A357879.
A122768 counts distinct submultisets of partitions.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A108917, A235130, A237194, A300061.

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

Mathematica

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 *)

Python

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

Formula

See A213074 for Metropolis and Stein's formulas.

a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n).

a(n) = A276107(2*n). - Max Alekseyev, Oct 17 2022

Extensions

Better description from Vladeta Jovovic, Mar 06 2000

More terms from Christian G. Bower, Oct 12 2001

Edited by N. J. A. Sloane, Jun 03 2012

More terms from Alois P. Heinz, Jul 10 2012

A237258 Number of strict partitions of 2n that include a partition of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 32, 43, 63, 84, 122, 158, 220, 293, 393, 511, 685, 881, 1156, 1485, 1925, 2445, 3147, 3952, 5019, 6323, 7924, 9862, 12336, 15259, 18900, 23294, 28646, 35091, 42985, 52341, 63694, 77336, 93588, 112973, 136367, 163874, 196638

Offset: 0

Clark Kimberling, Feb 05 2014

nonn

A strict partition is a partition into distinct parts.

			a(5) counts these partitions of 10: [5,4,1], [5,3,2], [4,3,2,1].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Cf. A000009, A237194, A235130.
The non-strict version is A002219, ranked by A357976.
These partitions are ranked by A357854.
A000712 counts distinct submultisets of partitions, strict A032302.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A006827, A064914, A108917, A276107, A300061, A357879.

z = 24; Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &,  Select[IntegerPartitions[2 nn], # == DeleteDuplicates[#] &]]; Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], nn]] >= 1 &]]], {nn, z}] (* Peter J. C. Moses, Feb 04 2014 *)

a(n) = A237194(2n,n).

a(31)-a(47) from Alois P. Heinz, Feb 07 2014

A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448

Offset: 1

Gus Wiseman, Oct 26 2022

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: {}
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

The partitions with these Heinz numbers are counted by A002219.
A subset of A300061.
The squarefree case is A357854, counted by A237258.
Positions of nonzero terms in A357879.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.

filter:= proc(n) local F,s,t,i,R;
  F:= ifactors(n)[2];
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], F);
  s:= add(t[1]*t[2],t=F)/2;
  if not s::integer then return false fi;
  try
  R:= Optimization:-Maximize(0, [add(F[i][1]*x[i],i=1..nops(F)) = s, seq(x[i]<= F[i][2],i=1..nops(F))], assume=nonnegint, depthlimit=20);
  catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
  end try;
  true
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Oct 26 2023

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 30, 70, 154, 165, 210, 273, 286, 390, 442, 462, 561, 595, 646, 714, 741, 858, 874, 910, 1045, 1155, 1173, 1254, 1326, 1330, 1334, 1495, 1653, 1771, 1794, 1798, 1870, 1938, 2139, 2145, 2294, 2415, 2465, 2470, 2530, 2622, 2639, 2730, 2926, 2945, 2958, 3034

Offset: 1

Gus Wiseman, Oct 27 2022

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: {}
    30: {1,2,3}
    70: {1,3,4}
   154: {1,4,5}
   165: {2,3,5}
   210: {1,2,3,4}
   273: {2,4,6}
   286: {1,5,6}
   390: {1,2,3,6}
For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.

The partitions with these Heinz numbers are counted by A237258.
A subset of A319241, squarefree case of A300061.
Squarefree positions of nonzero terms in A357879.
This is the squarefree case of A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[1000],SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Nov 07 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)

Row sums w/o the first column are A002219, ranks A357976, strict A237258.
Column k = 0 is A006827.
Row sums are A058696.
Column k = 1 is A108917.
The corresponding rank statistic is A357879 (without empty rows).
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, ranks A366321.
A182616 counts partitions of 2n with at least one odd part, ranks A366530.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sums of partitions, rank statistic A299701.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.

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]}]

T(n,1) = A108917(n).

A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Oct 27 2022

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 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}

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of nonzero terms are A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213074, A235130, A237258, A276107, A300061, A321144, A357975.

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

a(n) = Sum_{d|n} [A056239(d) = A056239(n/d)], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A236429 Number of representations of 0 as a sum of numbers d*k with d in {-1,1} and k in {1,2,...,n}, where the sum of the numbers k is 2n.

Original entry on oeis.org

1, 3, 6, 14, 25, 52, 86, 160, 260, 443, 688, 1146, 1721, 2716, 4040, 6176, 8975, 13482, 19218, 28167, 39799, 57081, 79503, 112987, 155368

Offset: 1

Clark Kimberling, Jan 25 2014

a(n) = number of partitions of 2n that contain a partition of n.

			a(3) counts these 6 representations of 0: 3-3, 3-2-1, 3-1-1-1, 2+1-2-1, 2+1-1-1-1, 1+1+1-1-1-1.

Cf. A236430, A235130.

p[n_] := p[n] = IntegerPartitions[n]; Map[({p1 = p[#], p2 = p[2 #]} &[#];    Length[Cases[p2, Apply[Alternatives, Map[Flatten[{_, #, _}] &, p1]]]]) &,  Range[12]]
Map[({p1 = p[# + 1], p2 = p[2 # + 1]} &[#]; Length[Cases[p2, Apply[Alternatives, Map[Flatten[{_, #, _}] &, p1]]]]) &, Range[12]]
(* Peter J. C. Moses, Jan 04 2014 *)

A236430 Number of representations of 1 as a sum of numbers d*k with d in {-1,1} and k in {1,2,...,n}, where the sum of the numbers k is 2n + 1.

Original entry on oeis.org

2, 5, 11, 22, 42, 76, 134, 228, 379, 606, 985, 1528, 2364, 3576, 5419, 7988, 11868, 17163, 24937, 35599, 50787, 71290, 100748, 139734, 194113

Offset: 1

Clark Kimberling, Jan 25 2014

a(n) = number of partitions of 2n+1 that contain a partition of n+1.

			a(2) counts these 5 representations of 1: 3-2, 3-1-1, 2+1-2, 2+1-1-1, 1+1+1-1-1.

Cf. A236429, A235130.

p[n_] := p[n] = IntegerPartitions[n]; Map[({p1 = p[#], p2 = p[2 #]} &[#];    Length[Cases[p2, Apply[Alternatives, Map[Flatten[{_, #, _}] &, p1]]]]) &,  Range[12]]
Map[({p1 = p[# + 1], p2 = p[2 # + 1]} &[#]; Length[Cases[p2, Apply[Alternatives, Map[Flatten[{_, #, _}] &, p1]]]]) &, Range[12]]
(* Peter J. C. Moses, Jan 04 2014 *)

cp's OEIS Frontend

A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.

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A237258 Number of strict partitions of 2n that include a partition of n.

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A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

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A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

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A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

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A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

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A236429 Number of representations of 0 as a sum of numbers d*k with d in {-1,1} and k in {1,2,...,n}, where the sum of the numbers k is 2n.

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A236430 Number of representations of 1 as a sum of numbers d*k with d in {-1,1} and k in {1,2,...,n}, where the sum of the numbers k is 2n + 1.

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