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 1-10 of 43 results. Next

A058360 Number of partitions of n whose reciprocal sum is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 19, 23, 25, 31, 33, 38, 42, 51, 57, 66, 75, 86, 97, 109, 122, 138, 155, 177, 200, 230, 253, 287, 320, 363, 405, 456, 507, 572, 639, 707, 785, 877, 971, 1079, 1198, 1334, 1476, 1635, 1802, 2002, 2213, 2445, 2700
Offset: 1

Views

Author

Robert G. Wilson v, Jan 25 2002, Sep 30 2009

Keywords

Comments

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

Examples

			a(12) = 7 because the partitions of 12 whose reciprocal sum is an integer are: {{6, 3, 2, 1}, {4, 4, 2, 1, 1}, {3, 3, 3, 1, 1, 1}, {2, 2, 2, 2, 2, 2}, {2, 2, 2, 2, 1, 1, 1, 1}, {2, 2, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}. Individually their reciprocal sums are: 2, 3, 4, 3, 6, 9 and 12.

References

  • From a question posted to the news group comp.soft-sys.math.mathematica by "Juan" (erfa11(AT)hotmail.com) at Steven M. Christensen and Associates, Inc. and MathTensor, Inc. Jan 22, 2002 08:46:57 +0000 (UTC).

Links

Crossrefs

Cf. A066824, A051908.

Programs

  • Mathematica
    (* first do *) << "Combinatorica`"; (* then *) f[n_] := Block[{c = i = 0, k = PartitionsP@n, p = {n}}, While[i < k, If[ IntegerQ[ Plus @@ (1/p)], c++ ]; i++; p = NextPartition@ p]; c]; Array[f, 61]
Table[Count[IntegerPartitions[n],?(IntegerQ[Total[1/#]]&)],{n,70}] (* _Harvey P. Dale, Sep 10 2022 *)
  • PARI
    a(n)=my(s); forpart(v=n,if(type(sum(i=1,#v,1/v[i]))=="t_INT",s++)); s \\ Charles R Greathouse IV, Dec 15 2020
  • PARI
    b(n)=if(n<4,return(n==0)); my(s); forpart(v=n, if(type(sum(i=1, #v, 1/v[i]))=="t_INT", s++), [2,n]); s
a(n)=my(s=1); vector(n,i,s+=b(i)) \\ Charles R Greathouse IV, Dec 15 2020

A316854 Number of integer partitions of n whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 1, 4, 2, 4, 1, 5, 1, 5, 1, 3, 4, 2, 5, 6, 5, 5, 4, 5, 5, 4, 8, 10, 9, 7, 5, 9, 10, 6, 12, 10, 8, 7, 6, 9, 13, 15, 8, 19, 13, 19, 19, 19, 18, 22, 26, 28, 28, 29, 22, 33, 29, 28, 38, 34, 26, 40, 32, 43, 39, 51, 38, 62, 46
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

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

Examples

			The a(36) = 10 partitions:
  (36),
  (30,6), (24,12), (18,18),
  (12,12,12),
  (12,12,6,6),
  (15,10,4,4,3), (12,12,6,3,3), (12,8,8,6,2),
  (6,6,6,6,6,6).

Links

Crossrefs

Cf. A000041, A051908, A058360, A316855, A316856, A316857.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[1/Sum[1/m,{m,#}]]&]],{n,30}]
ric[n_, p_, s_] := If[n == 0, If[IntegerQ[1/s], 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, 80] (* after Giovanni Resta in A316898, Robert G. Wilson v, Jul 23 2018 *)
  • PARI
    a(n)={my(s=0); forpart(p=n, if(frac(1/sum(i=1, #p, 1/p[i]))==0, s++)); s} \\ Andrew Howroyd, Jul 15 2018

Extensions

a(51)-a(77) from Giovanni Resta, Jul 15 2018

A316855 Heinz numbers of integer partitions whose reciprocal sum is 1.

Original entry on oeis.org

2, 9, 125, 147, 195, 2401, 3185, 4225, 6475, 6591, 7581, 10101, 10527, 16401, 20445, 20535, 21045, 25365, 46155, 107653, 123823, 142805, 161051, 164255, 164983, 171941, 218855, 228085, 267883, 304175, 312785, 333925, 333935, 335405, 343735, 355355, 390963
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

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).

Examples

			Sequence of all integer partitions whose reciprocal sum is 1 begins: (1), (2,2), (3,3,3), (4,4,2), (6,3,2), (4,4,4,4), (6,4,4,3), (6,6,3,3), (12,4,3,3), (6,6,6,2), (8,8,4,2).

Crossrefs

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316856, A316857.

Programs

  • Mathematica
    Select[Range[2,10000],Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]==1&]

A316857 Heinz numbers of integer partitions whose reciprocal sum 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, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 147, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

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).

Crossrefs

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316855, A316856.

Programs

  • Mathematica
    Select[Range[2,100],IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]&]

A316856 Heinz numbers of integer partitions whose reciprocal sum is an integer.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 147, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 729, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1323, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2401, 2592, 2646, 2916, 3120
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

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).

Examples

			195 is the Heinz number of (6,3,2), which has reciprocal sum 1/6 + 1/3 + 1/2 = 1, which is an integer, so 195 belongs to the sequence.
The sequence of all integer partitions whose reciprocal sum is an integer begins: (), (1), (11), (111), (22), (1111), (221), (11111), (2211), (111111), (22111), (2222).

Crossrefs

Cf. A000041, A051908, A056239, A058360, A072411, A296150, A316854, A316855, A316857.

Programs

  • Mathematica
    Select[Range[1000],IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,If[#==1,{},FactorInteger[#]]}]]&]

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

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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]&]

A316904 Heinz numbers of aperiodic integer partitions into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

2, 18, 72, 162, 195, 250, 288, 294, 390, 500, 588, 648, 780, 1125, 1152, 1176, 1458, 1560, 1755, 2000, 2250, 2352, 2592, 2646, 3120, 3185, 3510, 4000, 4500, 4608, 4704, 4802, 5292, 6240, 6370, 6475, 7020, 8450, 9000, 9408, 10125, 10368, 10527, 10584, 12480
Offset: 1

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Examples

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (221), (22111), (22221), (632), (3331), (2211111), (4421), (6321), (33311), (44211), (2222111).

Links

Crossrefs

Cf. A000837, A002966, A051908, A058360, A100953, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Programs

  • Mathematica
    Select[Range[2,20000],And[GCD@@FactorInteger[#][[All,2]]==1,GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A325618 Numbers m such that there exists an integer partition of m whose reciprocal factorial sum is 1.

Original entry on oeis.org

1, 4, 11, 18, 24, 31, 37, 44, 45, 50, 52, 57, 58, 65, 66, 70, 71, 73, 76, 78, 79, 83, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 107, 108, 109, 110, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 130, 131
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.
Conjecture: 137 is the greatest integer not in this sequence. - Charlie Neder, May 14 2019

Examples

			The sequence of terms together with an integer partition of each whose reciprocal factorial sum is 1 begins:
   1: (1)
   4: (2,2)
  11: (3,3,3,2)
  18: (3,3,3,3,3,3)
  24: (4,4,4,4,3,3,2)
  31: (4,4,4,4,3,3,3,3,3)
  37: (4,4,4,4,4,4,4,4,3,2)
  44: (4,4,4,4,4,4,4,4,3,3,3,3)
  45: (5,5,5,5,5,4,4,4,3,3,2)
  50: (4,4,4,4,4,4,4,4,4,4,4,4,2)

Links

Crossrefs

Factorial numbers: A000142, A002982, A007489, A011371, A022559, A064986, A115627, A284605, A325616.
Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316855, A325619, A325620, A325621, A325622, A325623, A325624.

Extensions

a(11)-a(55) from Charlie Neder, May 14 2019

A325619 Heinz numbers of integer partitions whose reciprocal factorial sum is 1.

Original entry on oeis.org

2, 9, 375, 15625
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      9: {2,2}
    375: {2,3,3,3}
  15625: {3,3,3,3,3,3}

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325624.

Programs

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

Formula

Contains prime(n)^(n!) for all n > 0, including 191581231380566414401 for n = 4.

A325620 Number of integer partitions of n whose reciprocal factorial sum is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 14, 14, 15, 16, 18, 19, 20, 22, 24, 25, 26, 28, 31, 33, 34, 36, 39, 41, 43, 45, 49, 52, 54, 57, 61, 65, 68, 71, 76, 80, 84, 88, 93, 98, 103, 107, 113
Offset: 1

Views

Author

Gus Wiseman, May 13 2019

Keywords

Comments

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Examples

			The initial terms count the following partitions:
  1: (1)
  2: (1,1)
  3: (1,1,1)
  4: (2,2)
  4: (1,1,1,1)
  5: (2,2,1)
  5: (1,1,1,1,1)
  6: (2,2,1,1)
  6: (1,1,1,1,1,1)
  7: (2,2,1,1,1)
  7: (1,1,1,1,1,1,1)
  8: (2,2,2,2)
  8: (2,2,1,1,1,1)
  8: (1,1,1,1,1,1,1,1)
  9: (2,2,2,2,1)
  9: (2,2,1,1,1,1,1)
  9: (1,1,1,1,1,1,1,1,1)

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.
Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316856, A325618, A325621, A325622.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],IntegerQ[Total[1/(#!)]]&]],{n,30}]
Showing 1-10 of 43 results. Next

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