A058360 -id:A058360 - cp's OEIS frontend

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

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

A339627 Number of partitions of n into distinct parts such that the sum of reciprocals of parts is an integer.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 4, 5, 4, 7, 4, 2, 2, 6, 6, 1, 3, 3, 4, 3, 4

Offset: 0

Ilya Gutkovskiy, Dec 10 2020

nonn

Also the number of ways to express an integer as the sum of distinct unit fractions such that the sum of the denominators is n.

			a(31) = 2 because we have 2 + 4 + 5 + 20 = 1 + 2 + 3 + 10 + 15 = 31 and 1/2 + 1/4 + 1/5 + 1/20 = 1, 1/1 + 1/2 + 1/3 + 1/10 + 1/15 = 2 are integers.

Cf. A051907, A051908, A058360, A339628, A339629.

A339628 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is an integer.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 120, 0, 0, 0, 0, 24, 144, 144, 120, 0, 0, 0, 24, 240, 720, 0, 0, 0, 120, 720, 240, 1440, 0, 0, 0, 840, 5760, 720, 5160, 744, 240, 720, 1440, 10080, 720, 5760, 5160, 1560, 5760, 2280, 16560, 6000, 7560

Offset: 0

Ilya Gutkovskiy, Dec 10 2020

nonn

Also the number of ordered ways to express an integer as the sum of distinct unit fractions such that the sum of the denominators is n.

			a(11) = 6 because we have 2 + 3 + 6 = 2 + 6 + 3 = 3 + 2 + 6 = 3 + 6 + 2 = 6 + 2 + 3 = 6 + 3 + 2 = 11 and 1/2 + 1/3 + 1/6 = 1 is an integer.

Cf. A051907, A051908, A058360, A339627, A339629.

A339629 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 24, 24, 24, 0, 0, 0, 0, 24, 120, 0, 0, 0, 0, 120, 0, 240, 0, 0, 0, 0, 840, 0, 720, 120, 24, 120, 0, 1440, 0, 720, 720, 120, 840, 0, 2280, 720, 960, 1080, 0, 840, 0, 11760, 0, 5040, 1440, 720

Offset: 0

Ilya Gutkovskiy, Dec 10 2020

nonn

Also the number of ordered ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is n.

			a(11) = 6 because we have 2 + 3 + 6 = 2 + 6 + 3 = 3 + 2 + 6 = 3 + 6 + 2 = 6 + 2 + 3 = 6 + 3 + 2 = 11 and 1/2 + 1/3 + 1/6 = 1.

Cf. A051907, A051908, A058360, A339627, A339628.

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

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

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A339627 Number of partitions of n into distinct parts such that the sum of reciprocals of parts is an integer.

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A339628 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is an integer.

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A339629 Number of compositions (ordered partitions) of n into distinct parts such that the sum of reciprocals of parts is 1.

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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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Examples

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Programs

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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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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

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