cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Gus Wiseman, Oct 01 2018

Keywords

Examples

			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)

Crossrefs

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319911, A319913.

Programs

  • Mathematica
    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}]

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.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Oct 01 2018

Keywords

Examples

			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)

Crossrefs

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319912, A319913.

Programs

  • Mathematica
    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}]

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

Views

Author

Gus Wiseman, Oct 01 2018

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Formula

a(n) >= A001055(n).
a(n) >= A002865(n).

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

Views

Author

Gus Wiseman, Oct 01 2018

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A000792, A001970, A005520, A048249, A066739, A070960, A201163, A275870, A319850, A318949, A319855, A319856, A319909, A319912, A319913.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Oct 17 2022

Keywords

Comments

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.

Examples

			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)

Crossrefs

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.

Programs

  • Mathematica
    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}]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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