A318585 -id:A318585 - cp's OEIS frontend

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

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

Gus Wiseman, Aug 29 2018

nonn

			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)

Giovanni Resta, Table of n, a(n) for n = 1..150

Cf. A000041, A051908, A058360, A289506, A289507, A316854, A316857.

Cf. A318584, A318585, A318587, A318588, A318589.

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

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

A318584 Number of integer partitions of n whose sum of reciprocals squared is 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 0, 0, 2, 0, 2, 1, 2, 2, 2, 1, 1, 2, 3, 0, 1, 1, 6, 2, 3, 2, 6, 2, 2, 3, 2, 6, 7, 2, 4, 3, 9, 4, 7, 5, 8, 8, 7, 9, 9, 11, 12, 7, 9, 11, 17, 9, 13, 12, 17, 16, 13, 15, 20, 26, 27, 18, 23

Offset: 0

Gus Wiseman, Aug 29 2018

nonn

The a(16) = 1 integer partition:
  (6,3,3,2,2,2)
The a(48) = 2 integer partitions:
  (18,9,9,3,3,2,2,2)
  (6,6,6,6,3,3,3,3,3,3,3,3)
The a(56) = 3 integer partitions:
  (12,6,6,4,4,4,4,4,4,4,2,2)
  (10,6,5,5,5,5,5,5,3,3,2,2)
  (6,6,4,4,4,4,4,4,4,4,3,3,3,3)
The a(60) = 6 integer partitions:
  (12,12,12,12,3,3,2,2,2)
  (8,8,8,8,6,4,4,4,3,3,2,2)
  (6,6,6,6,6,6,6,6,6,2,2,2)
  (12,12,12,4,3,3,3,3,3,3,2)
  (10,5,5,5,5,5,5,4,4,4,4,2,2)
  (6,4,4,4,4,4,4,4,4,4,4,4,4,3,3)

Cf. A000041, A051908, A058360, A289506, A289507, A316854, A316855.

Cf. A318585, A318586, A318587, A318588, A318589.

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

a(61)-a(100) from Alois P. Heinz, Aug 30 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

Gus Wiseman, Aug 29 2018

nonn

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

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

Cf. A001222, A051908, A056239, A112798, A289506, A289507, A316855, A316856, A316857.

Cf. A318584, A318585, A318586, A318588, A318589.

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

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

Gus Wiseman, Aug 29 2018

nonn

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

			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).

Cf. A001222, A056239, A058360, A112798, A289506, A289507, A316855, A316857.

Cf. A318584, A318585, A318586, A318587, A318589.

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

Gus Wiseman, Aug 29 2018

nonn

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

Cf. A001222, A051908, A056239, A058360, A112798, A289506, A289507, A316854, A316855.

Cf. A318584, A318585, A318586, A318587, A318588.

Select[Range[2,1000],IntegerQ[1/Total[If[#==1,{},Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]^2]]]]&]

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

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A318584 Number of integer partitions of n whose sum of reciprocals squared is 1.

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

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

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Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica