A066739 -id:A066739 - cp's OEIS frontend

A319911 Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n with no 1's, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.

0, 1, 1, 2, 3, 7, 9, 21, 31, 65, 102, 194, 321, 575, 956, 1652, 2684, 4576, 7367, 12035, 19490, 31185, 49418, 78595, 123393

Offset: 1

Gus Wiseman, Oct 01 2018

			The a(6) = 7 pairs:
  6 <= (6)
  6 <= (4,2)
  8 <= (4,2)
  6 <= (3,3)
  9 <= (3,3)
  6 <= (2,2,2)
  8 <= (2,2,2)
The a(7) = 9 pairs:
   7 <= (7)
   7 <= (5,2)
  10 <= (5,2)
   7 <= (4,3)
  12 <= (4,3)
   7 <= (3,2,2)
   8 <= (3,2,2)
  10 <= (3,2,2)
  12 <= (3,2,2)

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, 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/@Select[IntegerPartitions[n],FreeQ[#,1]&]],{n,30}]

A350029 Write n as n = k1 + k2 + ... + km, so that all k are distinct positive integers. a(n) is the maximum value of A001055(k1) + A001055(k2) + ... + A001055(km) over all such partitions.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25

Offset: 1

Thomas Scheuerle, Dec 09 2021

nonn

There exist cases where a(n) < a(n-1). Some examples are n = 53, 77, 113, 125, ...
There may exist multiple partitions of n = k1 + k2 + ... + km, where a(n) = A001055(k1) + A001055(k2) + ... + A001055(km). The number of such partitions is A350032(n).
It appears that a(n) - log(A066739(n)) > 0.
If the definition of this sequence would allow k1 = k2 = km, then this sequence would be the trivial sequence a(n) = n instead.

			  n = k1+k2+...+km   A001055(k1)+...+A001055(km)  = a(n)
--------------------------------------------------------
  1 = 1              1                            = 1
  2 = 2              1                            = 1
  3 = 1 + 2          1 + 1                        = 2
  4 = 1 + 3          1 + 1                        = 2
  5 = 1 + 4          1 + 2                        = 3
  6 = 1 + 2 + 3      1 + 1 + 1                    = 3

Thomas Scheuerle, Examples for n = 1 to 100.

Cf. A000009, A001055, A066739, A350032.

A350029(n, K=0) = { my(a=A001055(n)); while(n>2*K+=1, a=max(A001055(K)+A350029(n-K,K), a) ); a } \\ M. F. Hasler, Dec 09 2021

A319925 Number of integer partitions with no 1's whose parts can be combined together using additions and multiplications to obtain n.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 4, 10, 10, 18, 19, 38, 35, 62, 71, 113, 122, 199, 213, 329

Offset: 1

Gus Wiseman, Oct 01 2018

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

			The a(8) = 10 partitions (which are not all partitions of 8):
  (8),
  (42), (62), (53), (44),
  (222), (322), (422), (332),
  (2222).
For example, this list contains (322) because we can write 8 = 3*2+2.

Cf. A000792, A001970, A002865, A005520, A048249, A066739, A275870, A319850, A318949, A319909, A319910, A319912, A319913.

a(n) >= A001055(n).

a(n) >= A002865(n).

A350032 Write n as n = k1 + k2 + ... + km, so that all k are distinct positive integers and b = A001055(k1) + A001055(k2) + ... + A001055(km) becomes maximal. a(n) is the number of such partitions which attain this maximum.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 5, 2, 3, 1, 2, 3, 1, 2, 4, 1, 2, 5, 1, 2, 5, 1, 2, 5, 1, 2, 5, 1, 2, 4, 1, 1, 3, 7, 1, 3, 6, 1, 2, 5, 1, 1, 4, 1, 1, 3, 1, 11, 2, 4, 9, 2, 3, 7, 1, 2, 4, 7, 1, 3, 6, 1, 2, 4, 1, 1, 3, 1, 1, 2, 1, 6, 1, 2, 4, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1

Offset: 1

Thomas Scheuerle, Dec 09 2021

nonn

			a(6) = 2:
6 = 1 + 2 + 3 and A001055(1) + A001055(2) + A001055(3) = 3;
6 = 2 + 4     and A001055(2) + A001055(4) = 3.

Cf. A000009, A000041, A066739, A001055, A350029.

a(n) <= 1 + A066739(n) - A000041(n).

A066855 Triangle T(n,k) of numbers of representations of n as a sum of k products of positive integers, k=1..n. 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.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 4, 4, 2, 1, 1, 1, 5, 5, 4, 2, 1, 1, 3, 7, 8, 6, 4, 2, 1, 1, 2, 8, 11, 9, 6, 4, 2, 1, 1, 2, 11, 16, 14, 10, 6, 4, 2, 1, 1, 1, 11, 20, 20, 15, 10, 6, 4, 2, 1, 1, 4, 15, 28, 29, 23, 16, 10, 6, 4, 2, 1, 1, 1, 16, 33, 39, 33, 24, 16, 10, 6, 4, 2, 1, 1, 2, 19

Offset: 1

Vladeta Jovovic, Jan 21 2002

Row sums give A066739.

			[1], [1, 1], [1, 1, 1], [2, 2, 1, 1], [1, 3, 2, 1, 1], ... . 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, giving the batch [1, 3, 2, 1, 1].

Cf. A001055, A066739.

G.f.: Product_{m=1..infinity} (1-y*x^m)^(-A001055(m)). T(n, k) = Sum_{pi} Product_{m=1..n} binomial(p(m)+A001055(m)-1, p(m)), where pi runs through all nonnegative solutions of p(1)+2*p(2)+...+n*p(n)=n, p(1)+p(2)+...+p(n)=k.

A319907 Number of distinct 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 the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 01 2018

nonn

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

			The Heinz number of (3,3,2) is 75 and we have
    3+3+2 = 8,
    3+3*2 = 9,
    3*3+2 = 11,
  (3+3)*2 = 12,
  3*(3+2) = 15,
    3*3*2 = 18,
so a(75) = 6.

Cf. A000792, A001970, A005520, A048249, A066739, A070960, A201163, A275870, A319850, A318949, A319855, A319856, A319909, 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[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

A357858 Number of integer partitions that can be obtained by iteratively adding and multiplying together parts of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 6, 2, 3, 1, 7, 1, 3, 3, 11, 1, 7, 1, 8, 3, 3, 1, 14, 3, 3, 4, 8, 1, 11, 1, 19, 3, 3, 3, 18, 1, 3, 3, 18, 1, 12, 1, 8, 8, 3, 1, 27, 3, 10, 3, 8, 1, 16, 3, 19, 3, 3, 1, 25, 1, 3, 8, 33, 3, 12, 1, 8, 3, 12, 1, 35, 1, 3, 11, 8, 3, 12, 1, 34, 9

Offset: 1

Gus Wiseman, Oct 17 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The a(n) partitions for n = 1, 4, 8, 9, 12, 16, 20, 24:
  ()  (1)   (1)    (4)   (2)    (1)     (3)    (2)
      (2)   (2)    (22)  (3)    (2)     (4)    (3)
      (11)  (3)          (4)    (3)     (5)    (4)
            (11)         (21)   (4)     (6)    (5)
            (21)         (22)   (11)    (31)   (6)
            (111)        (31)   (21)    (32)   (21)
                         (211)  (22)    (41)   (22)
                                (31)    (311)  (31)
                                (111)          (32)
                                (211)          (41)
                                (1111)         (211)
                                               (221)
                                               (311)
                                               (2111)

The single-part partitions are counted by A319841, with an inverse A319913.
The minimum is A319855, maximum A319856.
A000041 counts integer partitions.
A001222 counts prime indices, distinct A001221.
A056239 adds up prime indices.
A066739 counts representations as a sum of products.
Cf. A000792, A001055, A001970, A005520, A048249, A063834, A066815, A318948, A319850, A319909, A319910.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
Table[Length[ReplaceListRepeated[{primeMS[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]]}]],{n,100}]

cp's OEIS Frontend

A319911 Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n with no 1's, 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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A350029 Write n as n = k1 + k2 + ... + km, so that all k are distinct positive integers. a(n) is the maximum value of A001055(k1) + A001055(k2) + ... + A001055(km) over all such partitions.

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PARI

A319925 Number of integer partitions with no 1's whose parts can be combined together using additions and multiplications to obtain n.

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A350032 Write n as n = k1 + k2 + ... + km, so that all k are distinct positive integers and b = A001055(k1) + A001055(k2) + ... + A001055(km) becomes maximal. a(n) is the number of such partitions which attain this maximum.

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A066855 Triangle T(n,k) of numbers of representations of n as a sum of k products of positive integers, k=1..n. 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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A319907 Number of distinct 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 the integer partition with Heinz number n.

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Mathematica

A357858 Number of integer partitions that can be obtained by iteratively adding and multiplying together parts of the integer partition with Heinz number n.

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