A318949 -id:A318949 - cp's OEIS frontend

A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent.

1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 518, 687, 960, 1267, 1747, 2294, 3106, 4052, 5449, 7063, 9365, 12092, 15914, 20422, 26639, 34029, 44091, 56076, 72110, 91306, 116808, 147272, 187224, 235201, 297594, 372390, 468844, 584644, 732942

Offset: 0

Naohiro Nomoto, Jan 16 2002

			For n=5, 5 = 4+1 = 2*2+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so a(5) = 8.
For n=8, 8 = 4*2 = 2*2*2 = ... = 4+4 = 2*2+4 = 2*2+2*2 = ...; note that there are 3 ways to factor the terms of 4+4. In general, if a partition contains a number k exactly r times, then the number of ways to factor the k's is the binomial coefficient C(A001055(k)+r-1,r).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000041, A001055.
Cf. A066815, A066816, A066806.
Cf. A001970, A050336, A063834, A065026, A066739, A281113, A284639, A318948, A318949.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60); # Alois P. Heinz, Apr 22 2012

p[ n_, 1 ] := If[ n==1, 1, 0 ]; p[ 1, k_ ] := 1; p[ n_, k_ ] := p[ n, k ]=p[ n, k-1 ]+If[ Mod[ n, k ]==0, p[ n/k, k ], 0 ]; A001055[ n_ ] := p[ n, n ]; a[ n_, 1 ] := 1; a[ 0, k_ ] := 1; a[ n_, k_ ] := If[ k>n, a[ n, n ], a[ n, k ]=a[ n, k-1 ]+Sum[ Binomial[ A001055[ k ]+r-1, r ]a[ n-k*r, k-1 ], {r, 1, Floor[ n/k ]} ] ]; a[ n_ ] := a[ n, n ]; (* p[ n, k ]=number of factorizations of n with factors <= k. a[ n, k ]=number of representations of n as a sum of products of positive integers, with summands <= k *)
b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; a[0] = 1; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#, #]&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)
facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]]],{n,50}] (* Gus Wiseman, Sep 05 2018 *)

from sympy.core.cache import cacheit
from sympy import divisors, isprime
@cacheit
def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum([0 if d>k else b(n//d, d) for d in divisors(n)[1:-1]]))
@cacheit
def a(n): return 1 if n==0 else sum(sum(d*b(d, d) for d in divisors(j))*a(n - j)  for j in range(1, n + 1))//n
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 19 2017, after Maple code

a(n) = Sum_{pi} Product_{m=1..n} binomial(k(m)+A001055(m)-1, k(m)), where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n. a(n)=1/n*Sum_{m=1..n} a(n-m)*b(m), n > 0, a(0)=1, b(m)=Sum_{d|m} d*A001055(d). Euler transform of A001055(n): Product_{m=1..infinity} (1-x^m)^(-A001055(m)). - Vladeta Jovovic, Jan 21 2002

Edited by Dean Hickerson, Jan 19 2002

A048249 Number of distinct values produced from sums and products of n unity arguments.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 11, 17, 23, 30, 44, 60, 80, 114, 156, 212, 296, 404, 556, 770, 1065, 1463, 2032, 2795, 3889, 5364, 7422, 10300, 14229, 19722, 27391, 37892, 52599, 73075, 101301, 140588, 195405, 271024, 376608, 523518, 726812, 1010576, 1405013, 1952498

Offset: 1

Tony Bartoletti

Values listed calculated by exhaustive search algorithm.
For n+1 operands (n operations) there are (2n)!/((n!)((n+1)!)) possible postfix forms over a single operator. For each such form, there are 2^n ways to assign 2 operators (here, sum and product). Calculate results and eliminate duplicates.
Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of an integer partition until only one part remains, starting with 1^n. - Gus Wiseman, Sep 29 2018

			a(3)=3 since (in postfix): 111** = 11*1* = 1, 111*+ = 11*1+ = 111+* = 11+1* = 2 and 111++ = 11+1+ = 3. Note that at n=7, the 11 possible values produced are the set {1,2,3,4,5,6,7,8,9,10,12}. This is the first n for which there are "skipped" values in the set.

Cf. A000792, A005520, A066739, A070960, A201163, A319850, A318949, A319855, A319856, A319909, A319910, A319911.

b:= proc(n) option remember; `if`(n=1, {1}, {seq(seq(seq(
     [f+g, f*g][], g=b(n-i)), f=b(i)), i=1..iquo(n, 2))})
    end:
a:= n-> nops(b(n)):
seq(a(n), n=1..35);  # Alois P. Heinz, May 05 2019

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
Table[Length[Select[ReplaceListRepeated[{Array[1&,n]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]],{n,10}] (* Gus Wiseman, Sep 29 2018 *)

from functools import cache
@cache
def f(m):
    if m == 1: return {1}
    out = set()
    for j in range(1, m//2+1):
        for x in f(j):
            for y in f(m-j):
                out.update([x + y, x * y])
    return out
def a(n): return len(f(n))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Aug 03 2022

Equals partial sum of "number of numbers of complexity n" (A005421). - Jonathan Vos Post, Apr 07 2006

More terms from David W. Wilson, Oct 10 2001

a(43)-a(44) from Alois P. Heinz, May 05 2019

A319910 Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.

Original entry on oeis.org

1, 3, 6, 11, 23, 48, 85, 178, 331, 619, 1176, 2183, 3876, 7013, 12447, 21719, 37628, 64570, 109723, 185055

Offset: 1

Gus Wiseman, Oct 01 2018

			The a(4) = 11 pairs:
  4 <= (4)
  3 <= (3,1)
  4 <= (3,1)
  4 <= (2,2)
  2 <= (2,1,1)
  3 <= (2,1,1)
  4 <= (2,1,1)
  1 <= (1,1,1,1)
  2 <= (1,1,1,1)
  3 <= (1,1,1,1)
  4 <= (1,1,1,1)

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319912, A319913.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
nexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Total[Length/@nexos/@IntegerPartitions[n]],{n,20}]

A319913 Number of distinct integer partitions whose parts can be combined together using additions and multiplications to obtain n, with the exception that 1's can only be added and not multiplied with other parts.

Original entry on oeis.org

1, 2, 3, 5, 7, 16, 20, 37, 53, 81, 107, 177, 227, 332, 449, 647, 830, 1162, 1480, 2032, 2597, 3447, 4348, 5775, 7251, 9374, 11758, 15026, 18640, 23688, 29220, 36771, 45128, 56168, 68674, 85015, 103394, 126923, 153871, 187911, 226653

Offset: 1

Gus Wiseman, Oct 01 2018

nonn

All parts of the integer partition must be used in such a combination.

			The a(7) = 20 partitions (which are not all partitions of 7):
  (7),
  (61), (52), (43),
  (511), (321), (421), (331), (322),
  (3111), (4111), (2211), (3211), (2221),
  (21111), (31111), (22111),
  (111111), (211111),
  (1111111).
This list contains (2211) because we can write 7 = (2+1)*2+1. It contains (321) because we can write 7 = 3*2+1, even though the sum of parts is only 6.

Charlie Neder, Table of n, a(n) for n = 1..46

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319912, A319925.

a(n) >= A000041(n).

a(n) >= A001055(n).

a(13)-a(41) from Charlie Neder, Jun 02 2019

A066815 Number of partitions of n into sums of products.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 33, 45, 69, 94, 148, 197, 289, 390, 575, 762, 1086, 1439, 2040, 2687, 3712, 4874, 6749, 8792, 11918, 15526, 20998, 27164, 36277, 46820, 62367, 80146, 105569, 135326, 177979, 227139, 296027, 377142, 490554, 622526, 804158

Offset: 0

Vladeta Jovovic, Jan 20 2002

nonn

Number of ways to choose a factorization of each part of an integer partition of n. - Gus Wiseman, Sep 05 2018

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = A001055(n). - Seiichi Manyama, Nov 14 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 14 partitions of 6 into sums of products:
  6, 2*3,
  5+1, 4+2, 2*2+2, 3+3,
  4+1+1, 2*2+1+1, 3+2+1, 2+2+2,
  3+1+1+1, 2+2+1+1,
  2+1+1+1+1,
  1+1+1+1+1+1.
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000041, A001055, A066739, A321460.

Cf. A001970, A050336, A063834, A065026, A281113, A284639, A318948, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]],{n,20}] (* Gus Wiseman, Sep 05 2018 *)

G.f.: Product_{k>=1} 1/(1-A001055(k)*x^k).

a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n > 0, a(0)=1, b(k)=Sum_{d|k} d*(A001055(d))^(k/d).

Renamed by T. D. Noe, May 24 2011

A319850 Number of distinct positive integers that can be obtained, starting with the initial interval partition (1, ..., n), by iteratively adding or multiplying together parts until only one part remains.

Original entry on oeis.org

1, 2, 5, 21, 94, 446, 2287, 12568, 78509

Offset: 1

Gus Wiseman, Sep 29 2018

			The n-th row lists all integers that can be obtained starting with (1, ..., n):
  1
  2 3
  5 6 7 8 9
  9 10 11 12 13 14 15 16 17 18 19 20 21 24 25 26 27 28 30 32 36

Cf. A000041, A001055, A001970, A048249, A066739, A066815, A070960, A201163, A318948, A318949, A319841.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
Table[Length[Select[ReplaceListRepeated[{Range[n]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]],{n,6}]

A325035 Product of sums of the multisets of prime indices of each prime index of 2 * n + 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 2, 4, 3, 2, 4, 4, 1, 4, 5, 3, 4, 4, 3, 6, 5, 2, 5, 4, 4, 4, 6, 3, 7, 5, 2, 6, 8, 4, 5, 6, 4, 6, 6, 1, 9, 8, 4, 5, 6, 5, 6, 6, 3, 7, 6, 4, 6, 10, 4, 6, 8, 3, 8, 9, 6, 8, 11, 5, 5, 6, 2, 7, 8, 5, 9, 8, 4, 7, 6, 4, 10, 12, 4, 8, 9, 6, 8, 9, 3

Offset: 0

Gus Wiseman, Mar 25 2019

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.

			91 has prime indices {4,6} with prime indices {{1,1},{1,2}} with sums {2,3} with product a(45) = 6.

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A318949.

Cf. A324926, A325031, A325032, A325033, A325034.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Plus@@@primeMS/@primeMS[n],{n,1,200,2}]

Fully multiplicative with a(prime(n)) = A056239(n), restricted to odd n.

A318948 Number of ways to choose an integer partition of each factor in a factorization of n.

Original entry on oeis.org

1, 2, 3, 9, 7, 17, 15, 40, 39, 56, 56, 126, 101, 165, 197, 336, 297, 496, 490, 774, 837, 1114, 1255, 1948, 2007, 2638, 3127, 4123, 4565, 6201, 6842, 9131, 10311, 12904, 14988, 19516, 21637, 26995, 31488, 39250, 44583, 55418, 63261, 77683, 89935, 108068, 124754

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

			The a(4) = 9 ways: (1+1)*(1+1), (1+1+1+1), (1+1)*(2), (2)*(1+1), (2+1+1), (2)*(2), (2+2), (3+1), (4).

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@PartitionsP/@fac,{fac,facs[n]}],{n,10}]

Dirichlet g.f.: Product_{n > 1} 1 / (1 - P(n) / n^s) where P = A000041. [clarified by Ilya Gutkovskiy, Oct 26 2019]

A267597 Number of sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of any submultiset of y is distinct.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 12, 14, 18, 23, 23, 32, 30, 35, 50, 48, 47, 56, 80, 77, 87, 105, 100, 134, 139, 145, 194, 170, 192, 250

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of product-sum knapsack partitions begins:
   0: ()
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5) (3,2)
   6: (6) (4,2) (3,3)
   7: (7) (5,2) (4,3)
   8: (8) (6,2) (5,3) (4,4)
   9: (9) (7,2) (6,3) (5,4)
  10: (10) (8,2) (7,3) (6,4) (5,5) (4,3,3)
  11: (11) (9,2) (8,3) (7,4) (6,5) (5,4,2) (5,3,3)
The partition (4,4,3) is not a sum-product knapsack partition of 11 because (4*4) = (4)+(4*3).
A complete list of all sums of products of multiset partitions of submultisets of (5,4,2) is:
            0 = 0
          (2) = 2
          (4) = 4
          (5) = 5
        (2*4) = 8
        (2*5) = 10
        (4*5) = 20
      (2*4*5) = 40
      (2)+(4) = 6
      (2)+(5) = 7
    (2)+(4*5) = 22
      (4)+(5) = 9
    (4)+(2*5) = 14
    (5)+(2*4) = 13
  (2)+(4)+(5) = 11
These are all distinct, so (5,4,2) is a sum-product knapsack partition of 11.

Sean A. Irvine, Java program (github)

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A320052, A320053, A320054, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rrtuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Plus,Apply[Times,Union@@mps/@Union[Subsets[q]],{2}],{1}]]];
Table[Length[rrtuks[n]],{n,12}]

a(13)-a(37) from Sean A. Irvine, Jul 13 2022

A319909 Number of distinct positive integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with 1^n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 4, 5, 10, 15, 21, 34, 49, 68, 101, 142, 197, 280, 387, 538, 751, 1045, 1442, 2010, 2772, 3865, 5339, 7396, 10273, 14201, 19693

Offset: 0

Gus Wiseman, Oct 01 2018

			We have
   7 = 1+1+1+1+1+1+1,
   8 = (1+1)*(1+1+1)+1+1,
   9 = (1+1)*(1+1)*(1+1)+1,
  10 = (1+1+1+1+1)*(1+1),
  12 = (1+1+1)*(1+1+1+1),
so a(7) = 5.

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A070960, A201163, A275870, A319850, A318948, A318949, A319912, A319913.

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
mexos[ptn_]:=If[Length[ptn]==0,{0},Union@@Select[ReplaceListRepeated[{Sort[ptn]},{{foe___,x_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x+y]],{foe___,x_?(#>1&),mie___,y_?(#>1&),afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];
Table[Length[mexos[Table[1,{n}]]],{n,30}]

cp's OEIS Frontend

A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent.

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A048249 Number of distinct values produced from sums and products of n unity arguments.

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A319910 Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.

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A319913 Number of distinct integer partitions whose parts can be combined together using additions and multiplications to obtain n, with the exception that 1's can only be added and not multiplied with other parts.

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A066815 Number of partitions of n into sums of products.

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A319850 Number of distinct positive integers that can be obtained, starting with the initial interval partition (1, ..., n), by iteratively adding or multiplying together parts until only one part remains.

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A325035 Product of sums of the multisets of prime indices of each prime index of 2 * n + 1.

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A318948 Number of ways to choose an integer partition of each factor in a factorization of n.

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