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.

Showing 1-5 of 5 results.

A318585 Number of integer partitions of n whose sum of reciprocals squared is an integer.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 12, 12, 13, 14, 16, 16, 18, 19, 21, 23, 26, 27, 29, 30, 34, 35, 39, 43, 48, 51, 55, 57, 63, 67, 74, 78, 84, 89, 99, 103, 112, 119, 132, 139, 148, 156, 170, 182, 199
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

From David A. Corneth, Sep 03 2018: (Start)
Let a valid tuple be a tuple of positive integers whose sum of reciprocals squared is an integer. Initially one only needs to consider tuples of positive integers where each element is > 1. After that some ones could be prepended to a valid tuple to find new valid tuples.
One could define a prime tuple as a valid tuple where no proper part with elements is a valid tuple. So (1) would be a prime tuple as no proper part of (1) has elements and is a valid tuple. Other examples of prime tuples are (2, 2, 2, 2) and (2, 2, 2, 3, 3, 6).
The list of distinct elements in a tuple could be whittled down by finding for each positive integer m the least sum of a prime tuple in which that integer is. For each m, that sum is at most m^3. (End)

Examples

			The a(26) = 7 integer partitions:
  (6332222222)
  (44442221111)
  (63322211111111)
  (22222222222211)
  (222222221111111111)
  (2222111111111111111111)
  (11111111111111111111111111)

Links

Crossrefs

Cf. A000041, A002865, A051908, A058360, A289506, A289507, A316854, A316856.
Cf. A316854, A318584, A318586, A318587, A318588, A318589.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[Total[#^(-2)]]&]],{n,30}]

Extensions

a(61)-a(70) from Giovanni Resta, Sep 03 2018

A318586 Number of integer partitions of n whose sum of reciprocals squared is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 2, 3, 3, 1, 4, 1, 3, 1, 2, 1, 5, 2, 1, 4, 5, 1, 5, 1, 6, 3, 2, 4, 8, 2, 4, 2, 6, 3, 9, 2, 4, 7, 5, 4, 11, 8, 7, 8, 9, 5, 12, 5, 16, 5, 10, 5, 25, 10, 9, 13, 18, 12, 18, 6, 11, 14, 22, 9, 24, 11, 21, 22, 25, 24, 23, 28, 32
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Examples

			The a(42) = 9 integer partitions:
  (42)
  (21,14,7)
  (18,9,9,6)
  (18,9,9,3,3)
  (20,10,4,4,4)
  (12,12,12,4,2)
  (10,5,5,5,5,5,5,2)
  (12,6,6,4,4,4,2,2,2)
  (6,6,4,4,4,4,3,3,3,3,2)

Links

Crossrefs

Cf. A000041, A051908, A058360, A289506, A289507, A316854, A316857.
Cf. A318584, A318585, A318587, A318588, A318589.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[1/Total[#^(-2)]]&]],{n,30}]

Extensions

a(61)-a(80) from Giovanni Resta, Sep 03 2018

A318587 Heinz numbers of integer partitions whose sum of reciprocals squared is 1.

Original entry on oeis.org

2, 81, 8775, 64827, 950625, 1953125, 7022925, 9055935, 21781575, 36020025, 50124555, 51883209, 57909033, 102984375, 118978125, 760816875, 816747435, 981059625, 1206902781, 1265058675, 1387132263, 2359670625, 3902169375, 4868424351, 5222768733, 5430160125
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

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

Examples

			The sequence of integer partitions with Heinz numbers in this sequence begins: (1), (2222), (633222), (4444222), (66333322).

Crossrefs

Cf. A001222, A051908, A056239, A112798, A289506, A289507, A316855, A316856, A316857.
Cf. A318584, A318585, A318586, A318588, A318589.

Programs

  • Mathematica
    Select[Range[100000],Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]==1&]

Extensions

a(6)-a(26) from Alois P. Heinz, Aug 30 2018

A318588 Heinz numbers of integer partitions whose sum of reciprocals squared is an integer.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 81, 128, 162, 256, 324, 512, 648, 1024, 1296, 2048, 2592, 4096, 5184, 6561, 8192, 8775, 10368, 13122, 16384, 17550, 20736, 26244, 32768, 35100, 41472, 52488, 64827, 65536, 70200, 82944, 104976, 129654, 131072, 140400, 165888, 209952
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

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

Examples

			Sequence of integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (11), (111), (1111), (11111), (111111), (2222), (1111111), (22221), (11111111), (222211), (111111111), (2222111).

Crossrefs

Cf. A001222, A056239, A058360, A112798, A289506, A289507, A316855, A316857.
Cf. A318584, A318585, A318586, A318587, A318589.

Programs

  • Mathematica
    Select[Range[10000],IntegerQ[Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]

A318589 Heinz numbers of integer partitions whose sum of reciprocals squared is the reciprocal of an integer.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

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

Crossrefs

Cf. A001222, A051908, A056239, A058360, A112798, A289506, A289507, A316854, A316855.
Cf. A318584, A318585, A318586, A318587, A318588.

Programs

  • Mathematica
    Select[Range[2,1000],IntegerQ[1/Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]
Showing 1-5 of 5 results.

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