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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A355389 Number of unordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276
Offset: 0

Views

Author

Gus Wiseman, Jul 04 2022

Keywords

Examples

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)

Crossrefs

The version for compositions is A006516.
Without distinctness we get A086737.
The unordered version is A355390, without distinctness A001255.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.

Programs

  • Maple
    a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024
  • Mathematica
    Table[Binomial[PartitionsP[n],2],{n,0,6}]
  • PARI
    a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

Formula

a(n) = binomial(A000041(n), 2) = A355390(n)/2.

A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

Original entry on oeis.org

1, 12, 36, 120, 180, 360, 840, 1260, 5400, 27000, 2520, 5040, 6300, 7560, 15120, 12600, 25200
Offset: 1

Views

Author

Gus Wiseman, Jul 02 2022

Keywords

Comments

The first position of -1 appears to be 18, pointed out by Amiram Eldar.
The terms are not always increasing.
The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.

Examples

			The terms together with their prime indices begin:
      1: {}
     12: {1,1,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    180: {1,1,2,2,3}
    360: {1,1,1,2,2,3}
    840: {1,1,1,2,3,4}
   1260: {1,1,2,2,3,4}
   5400: {1,1,1,2,2,2,3,3}
  27000: {1,1,1,2,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   5040: {1,1,1,1,2,2,3,4}
   6300: {1,1,2,2,3,3,4}
   7560: {1,1,1,2,2,2,3,4}
  15120: {1,1,1,1,2,2,2,3,4}
The terms together with their divisors satisfying the condition begin:
      1:   1
     12:   4,   6
     36:   4,   6,   9
    120:   8,  12,  20,  30
    180:  12,  18,  20,  30,  45
    360:   8,  12,  18,  20,  30,  45
    840:  24,  40,  56,  60,  84, 140, 210
   1260:  36,  60,  84,  90, 126, 140, 210, 315
   5400:   8,  12,  18,  20,  27,  30,  45,  50,  75
  27000:   8,  12,  18,  20,  27,  30,  45,  50,  75, 125
   2520:  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   5040:  16,  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   6300:  36,  60,  84,  90, 100, 126, 140, 150, 210, 225, 315, 350, 525

Crossrefs

These are the positions of first appearances in A355382, which is the version of A181591 without multiplicity.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 counts prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
A355383 counts cmpsbl. pairs of partitions with containment, comps. A355384.
Cf. A000712, A022811, A056239, A071625, A181819, A319910, A339006.

Programs

  • Mathematica
    tf=Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,1000}];
Table[Position[tf,n][[1,1]],{n,Select[Union[tf],SubsetQ[tf,Range[#]]&]}]

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).

A355390 Number of ordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 2, 6, 20, 42, 110, 210, 462, 870, 1722, 3080, 5852, 10100, 18090, 30800, 53130, 87912, 147840, 239610, 392502, 626472, 1003002, 1573770, 2479050, 3831806, 5931660, 9057090, 13819806, 20834660, 31399212, 46806122, 69697452, 102870306, 151523790, 221488806
Offset: 0

Views

Author

Gus Wiseman, Jul 04 2022

Keywords

Examples

			The a(0) = 0 through a(3) = 6 pairs:
  .  .  (11)(2)  (21)(3)
        (2)(11)  (3)(21)
                 (111)(3)
                 (3)(111)
                 (111)(21)
                 (21)(111)

Crossrefs

Without distinctness we have A001255, unordered A086737.
The version for compositions is A020522, unordered A006516.
The unordered version is A355389.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319910, A323433, A323583, A339006, A355385.

Programs

  • Mathematica
    Table[Length[Select[Tuples[IntegerPartitions[n],2],UnsameQ@@#&]],{n,0,15}]
  • PARI
    a(n) = 2*binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

Formula

a(n) = 2*A355389(n) = 2*binomial(A000041(n), 2).

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}]
Previous Showing 11-16 of 16 results.

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