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.

Previous Showing 11-16 of 16 results.

A316897 Number of 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, 5, 10, 9, 15, 10, 21, 12, 14, 16, 18, 9, 30, 18, 17, 17, 28, 16, 29, 26, 26, 30, 28, 33, 48, 31
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.

Examples

			The a(43) = 9 partitions:
(24,8,4,4,3)
(21,7,7,6,2)
(20,12,5,3,3)
(20,8,8,5,2)
(15,15,6,5,2)
(15,12,10,4,2)
(14,7,7,7,4,4)
(12,8,8,6,6,3)
(10,10,10,5,4,4).

Links

Crossrefs

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

Programs

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

Extensions

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

A316899 Number of integer partitions of n into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 6, 6, 8, 8, 10, 10, 14, 14, 19, 20, 25, 29, 33, 34, 41, 47, 54, 61, 75, 81, 97, 103, 121, 132, 155, 164, 200, 221, 252, 274, 320, 348, 405, 442, 501, 554, 639, 688, 784, 854, 968, 1053, 1198, 1298, 1475, 1602, 1797, 1965, 2213, 2399
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.

Examples

			The a(13) = 8 partitions are (63211), (442111), (33322), (3331111), (2222221), (222211111), (22111111111), (1111111111111).

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,IntegerQ[Sum[1/m,{m,#}]]]&]],{n,30}]
  • PARI
    a(n)={my(s=0); forpart(p=n, if(gcd(p)==1 && frac(sum(i=1, #p, 1/p[i]))==0, s++)); s} \\ Andrew Howroyd, Aug 26 2018

Extensions

a(51)-a(60) from Andrew Howroyd, Aug 26 2018

A316900 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is an integer.

Original entry on oeis.org

2, 4, 8, 16, 18, 32, 36, 64, 72, 128, 144, 162, 195, 250, 256, 288, 294, 324, 390, 500, 512, 576, 588, 648, 780, 1000, 1024, 1125, 1152, 1176, 1296, 1458, 1560, 1755, 2000, 2048, 2250, 2304, 2352, 2592, 2646, 2916, 3120, 3185, 3510, 4000, 4096, 4500, 4608
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).

Examples

			The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (11), (111), (1111), (221), (11111), (2211), (111111), (22111), (1111111), (221111), (22221), (632), (3331), (11111111).

Links

Crossrefs

Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Programs

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

A316901 Heinz numbers of integer partitions into relatively prime parts whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

2, 195, 3185, 5467, 6475, 6815, 8455, 10527, 15385, 16401, 17719, 20445, 20535, 21045, 25365, 28897, 40001, 46155, 49841, 50431, 54677, 92449, 101543, 113849, 123469, 137731, 156883, 164255, 171941, 185803, 218855, 228085, 230347, 261457, 267883, 274261
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).

Examples

			5467 is the Heinz number of (20,5,4) and 1/20 + 1/5 + 1/4 = 1/2, so 5467 belongs to the sequence.
The sequence of partitions whose Heinz numbers belong to this sequence begins: (1), (6,3,2), (6,4,4,3), (20,5,4), (12,4,3,3), (15,10,3), (24,8,3), (10,5,5,2)

Links

Crossrefs

Cf. A000837, A002966, A051908, A058360, A100953, A289509, A296150, A316854, A316855, A316856, A316857, A316888-A316904.

Programs

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

A316902 Heinz numbers of aperiodic integer partitions whose reciprocal sum is an integer.

Original entry on oeis.org

2, 18, 72, 147, 162, 195, 250, 288, 294, 390, 500, 588, 648, 780, 1125, 1152, 1176, 1323, 1458, 1560, 1755, 2000, 2250, 2352, 2592, 2646, 3120, 3185, 3510, 4000, 4500, 4608, 4704, 4802, 5292, 6240, 6370, 6475, 6591, 7020, 7581, 8450, 9000, 9408, 10101, 10125
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), (442), (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,IntegerQ[Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]

A316903 Heinz numbers of aperiodic integer partitions whose reciprocal sum is the reciprocal of an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 147, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257
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.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[2,1000],And[GCD@@FactorInteger[#][[All,2]]==1,IntegerQ[1/Sum[m[[2]]/PrimePi[m[[1]]],{m,FactorInteger[#]}]]]&]
Previous Showing 11-16 of 16 results.

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