A292886 -id:A292886 - cp's OEIS frontend

A196724 Number of subsets of {1..n} (including empty set) such that the pairwise products of distinct elements are all distinct.

1, 2, 4, 8, 16, 32, 58, 116, 212, 416, 720, 1440, 2340, 4680, 7920, 13024, 23328, 46656, 74168, 148336, 229856, 371424, 615304, 1230608, 1780224, 3401568, 5589360, 9468504, 14397744, 28795488, 40312128, 80624256, 131388480, 206363168, 335814288, 521401536

Offset: 0

Alois P. Heinz, Oct 06 2011

nonn

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product is A325860(n). - Gus Wiseman, Jun 03 2019

			a(6) = 58: from the 2^6=64 subsets of {1,2,3,4,5,6} only 6 do not have all the pairwise products of elements distinct: {1,2,3,6}, {2,3,4,6}, {1,2,3,4,6}, {1,2,3,5,6}, {2,3,4,5,6}, {1,2,3,4,5,6}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..50

Cf. A143823, A196719, A196720, A196721, A196722, A196723.
The subset case is A196724 (this sequence).
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Heinz numbers of the counterexamples are given by A325993.
Cf. A292886, A293627, A325860, A325861.

b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       sn[i]*sn[j], j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..20);

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] * sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]]; a[n_] := a[n] = b[n - 1, {n}] + If[n == 0, 0, a[n - 1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2017, translated from Maple *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Times@@@Subsets[#,{2}]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

Name edited by Gus Wiseman, Jun 03 2019

a(33)-a(35) from Fausto A. C. Cariboni, Oct 05 2020

A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 14, 2, 3, 4, 7, 1, 10, 1, 12, 3, 3, 3, 17, 1, 3, 3, 14, 1, 10, 1, 7, 7, 3, 1, 26, 2, 7, 3, 7, 1, 14, 3, 14, 3, 3, 1, 25, 1, 3, 7, 19, 3, 10, 1, 7, 3, 10, 1, 36, 1, 3, 7, 7, 3, 10, 1, 26, 7, 3

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

			The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
  ((2))  ((4))    ((6))    ((8))      ((12))     ((24))       ((30))
         ((2)*2)  ((2)*3)  ((2)*4)    ((2)*6)    ((3)*8)      ((5)*6)
                  (2*(3))  (2*(4))    (2*(6))    (3*(8))      (5*(6))
                           ((2)*2*2)  ((3)*4)    ((4)*6)      ((2)*15)
                                      (3*(4))    (4*(6))      (2*(15))
                                      ((2)*2*3)  ((2)*12)     ((3)*10)
                                      (2*2*(3))  (2*(12))     (3*(10))
                                                 ((2)*2*6)    ((2)*3*5)
                                                 (2*2*(6))    (2*(3)*5)
                                                 ((2)*3*4)    (2*3*(5))
                                                 (2*(3)*4)
                                                 (2*3*(4))
                                                 ((2)*2*2*3)
                                                 (2*2*2*(3))

The additive version is A000070 (strict: A015723).
The unpointed version is A001055 (strict: A045778, ordered: A074206, listed: A162247).
Allowing point (1) gives A057567.
Choosing a position instead of value gives A066637.
The ordered additive version is A336875.
A000005 counts divisors.
A001787 count normal multisets with a selected position.
A001792 counts compositions with a selected position.
A006128 counts partitions with a selected position.
A066186 count strongly normal multisets with a selected position.
A254577 counts ordered factorizations with a selected position.
Cf. A007716, A050336, A281113, A281116, A292886, A293627.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[fac]],{fac,facs[n]}],{n,50}]

a(n) = A057567(n) - A001055(n).

a(n) = Sum_{d|n, d>1} A001055(n/d).

A347460 Number of distinct possible alternating products of factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 5, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Oct 06 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			The a(n) alternating products for n = 1, 4, 8, 12, 24, 30, 36, 48, 60, 120:
  1  4  8    12   24   30    36   48    60    120
     1  2    3    6    10/3  9    12    15    30
        1/2  3/4  8/3  5/6   4    16/3  20/3  40/3
             1/3  2/3  3/10  1    3     15/4  15/2
                  3/8  2/15  4/9  3/4   12/5  24/5
                  1/6        1/4  1/3   3/5   10/3
                             1/9  3/16  5/12  5/6
                                  1/12  4/15  8/15
                                        3/20  3/10
                                        1/15  5/24
                                              2/15
                                              3/40
                                              1/30

Positions of 1's are 1 and A000040.
Positions of 2's appear to be A001358.
Positions of 3's appear to be A030078.
Dominates A038548, the version for reverse-alternating product.
Counting only integers gives A046951.
The even-length case is A072670.
The version for partitions (not factorizations) is A347461, reverse A347462.
The odd-length case is A347708.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A292886 counts knapsack factorizations, by sum A293627.
A299701 counts distinct subset-sums of prime indices, positive A304793.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
Cf. A002033, A119620, A143823, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@facs[n]]],{n,100}]

A301957 Number of distinct subset-products of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 5, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8

Offset: 1

Gus Wiseman, Mar 29 2018

nonn

A subset-product of an integer partition y is a product of some submultiset of y. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Number of distinct values obtained when A003963 is applied to all divisors of n. - Antti Karttunen, Sep 05 2018

			The distinct subset-products of (4,2,1,1) are 1, 2, 4, and 8, so a(84) = 4.
The distinct subset-products of (6,3,2) are 1, 2, 3, 6, 12, 18, and 36, so a(195) = 7.

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

Cf. A000712, A001055, A001227, A002865, A003963, A108917, A162247, A276024, A292886, A301854, A301855, A301856, A301970, A301979, A304793.

Table[If[n===1,1,Length[Union[Times@@@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

up_to = 65537;
A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
v003963 = vector(up_to,n,A003963(n));
A301957(n) = { my(m=Map(),s,k=0,c); fordiv(n,d,if(!mapisdefined(m,s = v003963[d],&c), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 05 2018

More terms from Antti Karttunen, Sep 05 2018

A293627 Number of knapsack factorizations whose factors sum to n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 19, 21, 27, 34, 45, 51, 69, 77, 100, 117, 146

Offset: 1

Gus Wiseman, Oct 23 2017

A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.

			The a(12) = 19 partitions are:
(12),
(10 2), (9 3), (8 4), (7 5), (6 6),
(8 2 2), (7 3 2), (6 4 2), (6 3 3), (5 5 2), (5 4 3), (4 4 4),
(6 2 2 2), (5 3 2 2), (4 3 3 2), (3 3 3 3),
(3 3 2 2 2),
(2 2 2 2 2 2).

Cf. A000041, A001055, A002033, A002865, A108917, A126796, A275972, A281116, A292886, A294150.

nn=22;
apsQ[y_]:=UnsameQ@@Times@@@Union[Rest@Subsets[y]];
Table[Length@Select[IntegerPartitions[n],apsQ],{n,nn}]

A347705 Number of factorizations of n with reverse-alternating product > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 8, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 12 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 2, 6, 8, 12, 24, 30, 48, 60:
  2   6     8       12      24        30      48          60
      2*3   2*4     2*6     3*8       5*6     6*8         2*30
            2*2*2   3*4     4*6       2*15    2*24        3*20
                    2*2*3   2*12      3*10    3*16        4*15
                            2*2*6     2*3*5   4*12        5*12
                            2*3*4             2*3*8       6*10
                            2*2*2*3           2*4*6       2*5*6
                                              3*4*4       3*4*5
                                              2*2*12      2*2*15
                                              2*2*2*6     2*3*10
                                              2*2*3*4     2*2*3*5
                                              2*2*2*2*3

Positions of 1's are A000430.
The weak version (>= instead of >) is A001055, non-reverse A347456.
The non-reverse version is A339890, strict A347447.
The version for reverse-alternating product 1 is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The even-length case is A347440, also the opposite reverse version.
Allowing any integer rev-alt product gives A347442, non-reverse A347437.
The version for partitions is A347449, non-reverse A347448.
A001055 counts factorizations (strict A045778, ordered A074206).
A038548 counts possible rev-alt products of factorizations, integer A046951.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A292886 counts knapsack factorizations, by sum A293627.
A347707 counts possible integer reverse-alternating products of partitions.
Cf. A028983, A119620, A339846, A347441, A347443, A347450, A347463, A347466.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[q[[-i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],revaltprod[#]>1&]],{n,100}]

a(n) = A001055(n) - A347438(n).

A325592 Triangle read by rows where T(n,k) is the number of length-k knapsack partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 3, 2, 0, 0, 1, 0, 1, 3, 4, 2, 0, 0, 1, 0, 1, 4, 3, 3, 0, 0, 0, 1, 0, 1, 4, 7, 2, 2, 0, 0, 0, 1, 0, 1, 5, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 5, 10, 6, 4, 2, 0, 0, 0, 0, 1, 0, 1, 6, 9, 5, 1, 2, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, May 15 2019

A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  0  1
  0  1  2  2  0  1
  0  1  3  2  0  0  1
  0  1  3  4  2  0  0  1
  0  1  4  3  3  0  0  0  1
  0  1  4  7  2  2  0  0  0  1
  0  1  5  6  4  2  0  0  0  0  1
  0  1  5 10  6  4  2  0  0  0  0  1
  0  1  6  9  5  1  2  0  0  0  0  0  1
  0  1  6 14 10  5  2  2  0  0  0  0  0  1
  0  1  7 13 11  3  3  2  0  0  0  0  0  0  1
  0  1  7 19 16  7  3  2  2  0  0  0  0  0  0  1
Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):
   (C)  (66)   (444)   (3333)  (81111)  (222222)  (111111111111)
        (75)   (543)   (5511)           (711111)
        (84)   (552)   (7221)
        (93)   (732)   (7311)
        (A2)   (741)   (9111)
        (B1)   (822)
               (831)
               (921)
               (A11)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010

Row sums are A000041.
Column k = 2 is A004526.
Column k = 3 is A325690.
Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.

Table[Length[Select[IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15},{k,0,n}]

A066637 Total number of elements in all factorizations of n with all factors > 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 8, 1, 3, 3, 12, 1, 8, 1, 8, 3, 3, 1, 17, 3, 3, 6, 8, 1, 10, 1, 20, 3, 3, 3, 22, 1, 3, 3, 17, 1, 10, 1, 8, 8, 3, 1, 34, 3, 8, 3, 8, 1, 17, 3, 17, 3, 3, 1, 27, 1, 3, 8, 35, 3, 10, 1, 8, 3, 10, 1, 46, 1, 3, 8, 8, 3, 10, 1, 34, 12, 3, 1, 27, 3, 3, 3, 17, 1, 27, 3, 8, 3, 3, 3

Offset: 1

Amarnath Murthy, Dec 28 2001

nonn

From Gus Wiseman, Apr 18 2021: (Start)
Number of ways to choose a factor index or position in a factorization of n. The version selecting a factor value is A339564. For example, the factorizations of n = 2, 4, 8, 12, 16, 24, 30 with a selected position (in parentheses) are:
  ((2))  ((4))    ((8))      ((12))     ((16))       ((24))       ((30))
         ((2)*2)  ((2)*4)    ((2)*6)    ((2)*8)      ((3)*8)      ((5)*6)
         (2*(2))  (2*(4))    (2*(6))    (2*(8))      (3*(8))      (5*(6))
                  ((2)*2*2)  ((3)*4)    ((4)*4)      ((4)*6)      ((2)*15)
                  (2*(2)*2)  (3*(4))    (4*(4))      (4*(6))      (2*(15))
                  (2*2*(2))  ((2)*2*3)  ((2)*2*4)    ((2)*12)     ((3)*10)
                             (2*(2)*3)  (2*(2)*4)    (2*(12))     (3*(10))
                             (2*2*(3))  (2*2*(4))    ((2)*2*6)    ((2)*3*5)
                                        ((2)*2*2*2)  (2*(2)*6)    (2*(3)*5)
                                        (2*(2)*2*2)  (2*2*(6))    (2*3*(5))
                                        (2*2*(2)*2)  ((2)*3*4)
                                        (2*2*2*(2))  (2*(3)*4)
                                                     (2*3*(4))
                                                     ((2)*2*2*3)
                                                     (2*(2)*2*3)
                                                     (2*2*(2)*3)
                                                     (2*2*2*(3))
(End)

			a(12) = 8: there are 4 factorizations of 12: (12), (6*2), (4*3), (3*2*2) having 1, 2, 2, 3 elements respectively, a total of 8.

Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy, Length and extent of Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

The version for normal multisets is A001787.
The version for compositions is A001792.
The version for partitions is A006128 (strict: A015723).
Choosing a value instead of position gives A339564.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A002033 and A074206 count ordered factorizations.
A067824 counts strict chains of divisors starting with n.
A336875 counts compositions with a selected part.
Cf. A000005, A000041, A045778, A050336, A066186, A162247, A264401, A281116, A292504, A292886, A322794.

# Return a list of lists which are factorizations (product representations)
# of n. Within each sublist, the factors are sorted. A minimum factor in
# each element of sublists returned can be specified with 'mincomp'.
# If mincomp=2, the number of sublists contained in the list returned is A001055(n).
# Example:
# n=8 and mincomp=2 return [[2,2,2],[4,8],[8]]
listProdRep := proc(n,mincomp)
    local dvs,resul,f,i,j,rli,tmp ;
    resul := [] ;
    # list returned is empty if n < mincomp
    if n >= mincomp then
        if n = 1 then
            RETURN([1]) ;
        else
            # compute the divisors, and take each divisor
            # as a head element (minimum element) of one of the
            # sublists. Example: for n=8 use {1,2,4,8}, and consider
            # (for mincomp=2) sublists [2,...], [4,...] and [8].
            dvs := numtheory[divisors](n) ;
            for i from 1 to nops(dvs) do
                # select the head element 'f' from the divisors
                f := op(i,dvs) ;
                # if this is already the maximum divisor n
                # itself, this head element is the last in
                # the sublist
                if f =n and f >= mincomp then
                    resul := [op(resul),[f]] ;
                elif f >= mincomp then
                    # if this is not the maximum element
                    # n itself, produce all factorizations
                    # of the remaining factor recursively.
                    rli := procname(n/f,f) ;
                    # Prepend all the results produced
                    # from the recursion with the head
                    # element for the result.
                    for j from 1 to nops(rli) do
                        tmp := [f,op(op(j,rli))] ;
                        resul := [op(resul),tmp] ;
                    od ;
                fi ;
            od ;
        fi ;
    fi ;
    resul ;
end:
A066637 := proc(n)
    local f,d;
    a := 0 ;
    for d in listProdRep(n,2) do
        a := a+nops(d) ;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 11 2013
# second Maple program:
with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, [1$2])+
      `if`(isprime(n), 0, (p-> p+[0, p[1]])(add(
      `if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n})))
    end:
a:= n-> `if`(n<2, 0, b(n$2)[2]):
seq(a(n), n=1..120); # Alois P. Heinz, Feb 12 2019

g[1, r_] := g[1, r]={1, 0}; g[n_, r_] := g[n, r]=Module[{ds, i, val}, ds=Select[Divisors[n], 1<#<=r&]; val={0, 0}+Sum[g[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]; val+{0, val[[1]]}]; a[n_] := g[n, n][[2]]; a/@Range[95] (* g[n, r] = {c, f}, where c is the number of factorizations of n with factors <= r and f is the total number of factors in them. - Dean Hickerson, Oct 28 2002 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];Table[Sum[Length[fac],{fac,facs[n]}],{n,50}] (* Gus Wiseman, Apr 18 2021 *)

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

A301856 Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 14, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 27, 4, 4, 4, 36, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 62, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 47, 4, 17, 1, 12, 4, 17, 1, 90, 1, 4, 12, 12, 4, 17

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

For a finite multiset p of positive integers greater than 1 with product n, a pair (t > 1, p) is defined to be a subset-product if there exists a nonempty submultiset of p with product t.

			The a(12) = 12 subset-products:
12<=(2*2*3), 6<=(2*2*3), 4<=(2*2*3), 3<=(2*2*3), 2<=(2*2*3),
12<=(2*6),   6<=(2*6),   4<=(3*4),   3<=(3*4),   2<=(2*6),
12<=(3*4),
12<=(12).
The a(16) = 14 subset-products:
16<=(16),
16<=(4*4),
16<=(2*8),     8<=(2*8),     4<=(4*4),     2<=(2*8),
16<=(2*2*4),   8<=(2*2*4),   4<=(2*2*4),   2<=(2*2*4),
16<=(2*2*2*2), 8<=(2*2*2*2), 4<=(2*2*2*2), 2<=(2*2*2*2).

Cf. A001055, A000712, A045778, A108917, A162247, A276024, A281116, A284640, A292886, A301829, A301830.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[Times@@@Rest[Subsets[f]]]],{f,facs[n]}],{n,100}]

cp's OEIS Frontend

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