A066815 -id:A066815 - 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

A318949 Number of ways to write n as an orderless product of orderless sums.

Original entry on oeis.org

1, 2, 3, 8, 7, 17, 15, 36, 36, 56, 56, 123, 101, 165, 197, 310, 297, 490, 490, 767, 837, 1114, 1255, 1925, 1986, 2638, 3110, 4108, 4565, 6201, 6842, 9043, 10311, 12904, 14988, 19398, 21637, 26995, 31488, 39180, 44583, 55418, 63261, 77627, 89914, 108068, 124754

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

			The a(6) = 17 ways:
  (6)              (2)*(3)
  (3+3)            (2)*(2+1)
  (4+2)            (2)*(1+1+1)
  (5+1)            (1+1)*(3)
  (2+2+2)          (1+1)*(2+1)
  (3+2+1)          (1+1)*(1+1+1)
  (4+1+1)
  (2+2+1+1)
  (3+1+1+1)
  (2+1+1+1+1)
  (1+1+1+1+1+1)

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A281113, A284639, A318948.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
prodsums[n_]:=Union[Sort/@Join@@Table[Tuples[IntegerPartitions/@fac],{fac,facs[n]}]];
Table[Length[prodsums[n]],{n,30}]

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={MultEulerT(vector(n, n, numbpart(n)))} \\ Andrew Howroyd, Oct 26 2019

Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^p(k), where p(k) = number of partitions of k (A000041). - Ilya Gutkovskiy, Oct 26 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

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}]

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]

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}]

A319912 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 any two or multiplying any two non-1 parts of y until only one part (equal to m) remains.

Original entry on oeis.org

1, 2, 3, 5, 12, 30, 53, 128, 247, 493, 989, 1889, 3434, 6390, 11526, 20400, 35818, 62083, 106223, 180170

Offset: 1

Gus Wiseman, Oct 01 2018

			The a(6) = 30 pairs:
  1 <= (1)
  2 <= (2)
  2 <= (1,1)
  3 <= (3)
  3 <= (2,1)
  3 <= (1,1,1)
  4 <= (4)
  4 <= (2,2)
  4 <= (3,1)
  4 <= (2,1,1)
  4 <= (1,1,1,1)
  5 <= (5)
  5 <= (3,2)
  5 <= (4,1)
  5 <= (2,2,1)
  5 <= (3,1,1)
  5 <= (2,1,1,1)
  5 <= (1,1,1,1,1)
  6 <= (6)
  6 <= (3,2)
  6 <= (3,3)
  6 <= (4,2)
  6 <= (5,1)
  6 <= (2,2,1)
  6 <= (2,2,2)
  6 <= (3,1,1)
  6 <= (3,2,1)
  6 <= (4,1,1)
  6 <= (2,1,1,1)
  6 <= (2,2,1,1)
  6 <= (3,1,1,1)
  6 <= (1,1,1,1,1)
  6 <= (2,1,1,1,1)
  6 <= (1,1,1,1,1,1)

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319911, 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[Total[Length/@mexos/@IntegerPartitions[n]],{n,20}]

A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 5, 1, 7, 4, 8, 3, 6, 5, 9, 2, 6, 6, 6, 4, 10, 5, 11, 1, 7, 7, 7, 4, 12, 8, 8, 3, 13, 6, 14, 5, 7, 9, 15, 2, 8, 6, 9, 6, 16, 6, 8, 4, 10, 10, 17, 5, 18, 11, 8, 1, 9, 7, 19, 7, 11, 7, 20, 4, 21, 12, 8, 8, 9, 8, 22, 3, 8

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(30) = 5 because the minimum number that can be obtained starting with (3,2,1) is 3+2*1 = 5.

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319856.

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[Min[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

a(1) = 0, a(n) = max(A056239(n) - A007814(n), 1). - Charlie Neder, Oct 03 2018

A319856 Maximum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 6, 4, 7, 6, 8, 6, 8, 6, 9, 6, 9, 7, 8, 8, 10, 9, 11, 6, 10, 8, 12, 9, 12, 9, 12, 9, 13, 12, 14, 10, 12, 10, 15, 9, 16, 12, 14, 12, 16, 12, 15, 12, 16, 11, 17, 12, 18, 12, 16, 9, 18, 15, 19, 14, 18, 16, 20, 12, 21, 13

Offset: 1

Gus Wiseman, Sep 29 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(30) = 9 because the maximum number that can be obtained starting with (3,2,1) is 3*(2+1) = 9.

Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319855.

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[Max[nexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

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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A318949 Number of ways to write n as an orderless product of orderless sums.

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

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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.

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A319912 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 any two or multiplying any two non-1 parts of y until only one part (equal to m) remains.

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A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

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