A316904 -id:A316904 - cp's OEIS frontend

A051908 Number of ways to express 1 as the sum of unit fractions such that the sum of the denominators is n.

1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 3, 0, 1, 1, 1, 1, 2, 3, 2, 2, 1, 2, 2, 2, 4, 5, 5, 2, 4, 5, 5, 9, 4, 4, 6, 4, 4, 7, 8, 4, 10, 9, 9, 11, 8, 13, 13, 15, 16, 21, 18, 16, 22, 19, 18, 30, 24, 19, 26, 28, 26, 29, 35, 29, 44, 28, 47, 48

Offset: 1

Jud McCranie, Dec 16 1999

nonn

Also the number of partitions of n whose reciprocal sums to 1; "exact partitions". - Robert G. Wilson v, Sep 30 2009

			1 = 1/2 + 1/2, the sum of denominators is 4, and this is the only expression of 1 as unit fractions with denominator sum 4, so a(4)=1.
The a(22) = 3 partitions whose reciprocal sum is 1 are (12,4,3,3), (10,5,5,2), (8,8,4,2). - _Gus Wiseman_, Jul 16 2018

Derrick Niederman, "Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed", a Perigee Book, Penguin Group, NY, 2009, pp. 82-83. [From Robert G. Wilson v, Sep 30 2009]

David A. Corneth, Table of n, a(n) for n = 1..200 (terms a(1)-a(86) from Jud McCranie, a(87)-a(88) from Robert G. Wilson v, a(89)-a(100) from Seiichi Manyama)
David A. Corneth, Tuples up to n = 170
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums
Index entries for sequences related to Egyptian fractions

A028229 lists n such that a(n)=0.

Cf. A002966, A058360, A270599, A316854, A316855, A316888-A316904.

(* first do *) << "Combinatorica`"; (* then *) f[n_] := Block[{c = i = 0, k = PartitionsP@n, p = {n}}, While[i < k, If[1 == Plus @@ (1/p), c++ ]; i++; p = NextPartition@p]; c]; Array[f, 88] (* Robert G. Wilson v, Sep 30 2009 *)
Table[Length[Select[IntegerPartitions[n],Sum[1/m,{m,#}]==1&]],{n,30}] (* Gus Wiseman, Jul 16 2018 *)

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A051908(n)
  ary = [1]
  (2..n).each{|m|
    cnt = 0
    partition(m, 2, m).each{|ary|
      cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1
    }
    ary << cnt
  }
  ary
end
p A051908(100) # Seiichi Manyama, May 31 2016

a(n) > 0 for n > 23.

A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

2, 195, 3185, 6475, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 164255, 171941, 218855, 228085, 267883, 312785, 333925, 333935, 335405, 343735, 355355, 414295, 442975, 474513, 527425, 549575, 607475, 633777, 691041, 711321, 722425, 753865, 804837, 822783

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.
Does not contain 29888089, which belongs to A316890 and is the Heinz number of a periodic partition.

			The partition (6,4,4,3) with Heinz number 3185 is aperiodic, has relatively prime parts, and 1/6 + 1/4 + 1/4 + 1/3 = 1, so 3185 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (6,3,2), (6,4,4,3), (12,4,3,3), (10,5,5,2), (20,5,4,2), (15,10,3,2), (12,12,3,2), (18,9,3,2), (24,8,3,2), (42,7,3,2).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A007916, A051908, A100953, A289509, A296150, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A316890 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

2, 195, 3185, 6475, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 164255, 171941, 218855, 228085, 267883, 312785, 333925, 333935, 335405, 343735, 355355, 414295, 442975, 474513, 527425, 549575, 607475, 633777, 691041, 711321, 722425, 753865, 804837, 822783

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Includes 29888089, which is the first perfect power in the sequence and is absent from A316888.

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 4, 1, 3, 1, 1, 1, 3, 1, 8, 3, 1, 1, 9, 2, 11, 3, 3, 3, 5, 2, 7, 6, 4, 7, 12, 5, 14, 6, 11, 12, 25, 11, 27, 17, 15, 19, 25, 9, 37, 20, 21, 19, 31, 19, 38, 33, 26, 37, 38, 36, 64, 39, 46, 53, 63, 39, 80, 63, 65, 66, 94, 59, 105

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

Records: 1, 2, 4, 8, 9, 11, 12, 14, 25, 27, 37, 38, 64, 80, 94, 105, 119, 154, 184, ..., . - Robert G. Wilson v, Jul 18 2018

			The a(37) = 8 partitions: (20,12,5), (15,12,10), (24,8,3,2), (15,10,6,6), (20,5,4,4,4), (15,10,6,3,3), (14,7,7,7,2), (10,10,10,5,2).

Robert G. Wilson v, Table of n, a(n) for n = 1..150
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888, A316889, A316890, A316891, A316892, A316893, A316904.

f[n_] := Block[{k = 1, s = 0, lmt = 1 + PartitionsP@ n}, While[k < lmt, s += Length[ Select[ IntegerPartitions[n, {k, k}], GCD @@ # == 1 && IntegerQ[1/Sum[1/m, {m, #}]] &]]; k++]; s]; Array[f, 50] (* slightly modified by Robert G. Wilson v, Jul 17 2018 *) (* or *)
ric[n_,p_,s_] := If[n==0, If[IntegerQ[1/s] && GCD @@ p == 1, c++], Do[ If[s + 1/i <= 1, ric[n-i, Append[p, i], s + 1/i]], {i, Min[p[[-1]], n], 1, -1}]]; a[n_] := (c=0; Do[ric[n-j, {j}, 1/j], {j, n}]; c); Array[a, 50] (* Giovanni Resta, Jul 18 2018 *)

a(51)-a(91) from Robert G. Wilson v, Jul 17 2018

A316889 Heinz numbers of aperiodic integer partitions whose reciprocal sum is 1.

Original entry on oeis.org

2, 147, 195, 3185, 6475, 6591, 7581, 10101, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 107653, 123823, 142805, 164255, 164983, 171941, 218855, 228085, 267883, 304175, 312785, 333925, 333935, 335405, 343735, 355355, 390963, 414295, 442975, 444925, 455975

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A partition is aperiodic if its multiplicities are relatively prime.

			Sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (4,4,2), (6,3,2), (6,4,4,3), (12,4,3,3), (6,6,6,2), (8,8,4,2), (12,6,4,2), (10,5,5,2), (20,5,4,2), (15,10,3,2), (12,12,3,2), (18,9,3,2), (24,8,3,2), (42,7,3,2).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A007916, A051908, A058360, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Select[Range[2,100000],And[GCD@@FactorInteger[#][[All,2]]==1,Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1]&]

A316891 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 5, 2, 7, 4, 7, 6, 13, 7, 18, 12, 20, 17, 32, 20, 39, 31, 47, 45, 74, 56, 96, 83, 109, 105, 151, 130, 199, 183, 234, 232, 319, 286, 404, 386, 473, 488, 638, 599, 782, 767, 931, 960, 1197, 1165, 1465, 1477, 1747, 1814, 2212, 2196

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

			The a(17) = 13 partitions:
(6443),
(44441),
(3332222), (6322211),
(44222111),
(222222221), (333221111), (632111111),
(4421111111),
(22222211111), (33311111111),
(2222111111111),
(221111111111111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316856, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]==1,IntegerQ[Sum[1/m,{m,#}]]]&]],{n,50}]

a(51)-a(60) from Alois P. Heinz, Jul 18 2018

A316893 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 3, 1, 2, 1, 1, 1, 2, 1, 5, 3, 1, 1, 5, 2, 9, 3, 3, 3, 4, 2, 6, 6, 3, 4, 9, 5, 10, 4, 10, 8, 15, 10, 21, 12, 14, 16, 18, 9, 30, 18, 17, 16, 28, 16, 29, 25, 26, 30, 28, 33, 48, 31

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]==1,Sum[1/m,{m,#}]==1]&]],{n,30}]

a(71)-a(80) from Giovanni Resta, Jul 16 2018

A316894 Number of aperiodic integer partitions of n whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 4, 1, 5, 1, 2, 3, 2, 4, 5, 5, 5, 4, 3, 5, 4, 8, 6, 9, 7, 5, 6, 10, 6, 12, 8, 7, 7, 6, 6, 12, 12, 8, 18, 13, 16, 19, 17, 18, 21, 26, 26, 28, 29, 21, 29, 29, 27, 38, 32, 26, 37, 32, 38, 39, 49, 36, 61, 46, 55

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316857, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Length/@Split[#]==1,IntegerQ[1/Sum[1/m,{m,#}]]]&]],{n,30}]

a(51)-a(78) from Giovanni Resta, Jul 16 2018

A316895 Number of aperiodic integer partitions of n whose reciprocal sum is an integer.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 2, 5, 2, 7, 5, 7, 6, 13, 8, 18, 13, 20, 19, 32, 21, 39, 35, 49, 48, 74, 60, 96, 86, 110, 111, 151, 135, 199, 192, 235, 239, 319, 299, 404, 394, 477, 506, 638, 609, 782, 788, 934, 978, 1197, 1193, 1466, 1501, 1752, 1851, 2212, 2227

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

			The a(11) = 5 partitions are (632), (4421), (33311), (2222111), (221111111).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Length/@Split[#]==1,IntegerQ[Sum[1/m,{m,#}]]]&]],{n,30}]

a(51)-a(60) from Alois P. Heinz, Jul 17 2018

A316896 Number of aperiodic integer partitions of n whose reciprocal sum is 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 3, 0, 1, 0, 1, 1, 2, 3, 2, 2, 1, 2, 2, 2, 3, 5, 5, 2, 2, 5, 5, 9, 3, 4, 6, 4, 3, 6, 8, 4, 10, 9, 8, 11, 7, 13, 12, 15, 15, 21, 18, 16, 21, 19, 17, 30, 23, 19, 23, 28, 25, 29, 34, 29, 44, 28, 46, 48, 42

Offset: 1

Gus Wiseman, Jul 16 2018

nonn

The reciprocal sum of (y_1, ..., y_k) is 1/y_1 + ... + 1/y_k.

A partition is aperiodic if its multiplicities are relatively prime.

			The a(37) = 5 partitions are (24,8,3,2), (20,5,4,4,4), (15,10,6,3,3), (14,7,7,7,2), (10,10,10,5,2).

Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Cf. A000837, A002966, A051908, A058360, A100953, A316854, A316888-A316904.

Table[Length[Select[IntegerPartitions[n],And[GCD@@Length/@Split[#]==1,Sum[1/m,{m,#}]==1]&]],{n,30}]

a(51)-a(80) from Giovanni Resta, Jul 16 2018

cp's OEIS Frontend

A051908 Number of ways to express 1 as the sum of unit fractions such that the sum of the denominators is n.

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A316888 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is 1.

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A316890 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is 1.

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A316898 Number of integer partitions of n into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

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A316889 Heinz numbers of aperiodic integer partitions whose reciprocal sum is 1.

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A316891 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is an integer.

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Mathematica

Extensions

A316893 Number of aperiodic integer partitions of n into relatively prime parts whose reciprocal sum is 1.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A316894 Number of aperiodic integer partitions of n whose reciprocal sum is the reciprocal of an integer.

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