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 31-40 of 43 results. Next

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

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

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Extensions

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

A168173 Number of partitions of n in which the sum of reciprocals of parts is less than 1.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 4, 6, 8, 12, 13, 16, 18, 21, 25, 32, 38, 46, 55, 65, 78, 92, 103, 122, 140, 165, 193, 229, 264, 305, 345, 395, 451, 517, 590, 682, 781, 893, 1013, 1165, 1324, 1518, 1717, 1945, 2188, 2468, 2753, 3089, 3457, 3865, 4321, 4856, 5441, 6108, 6831
Offset: 1

Views

Author

Vladeta Jovovic, Nov 19 2009

Keywords

Crossrefs

Cf. A051908.

Programs

  • Maple
    a := proc (n) local P, ct, j: with(combinat): P := partition(n): ct := 0: for j to numbpart(n) do if add(1/P[j][i], i = 1 .. nops(P[j])) < 1 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 60); # Emeric Deutsch, Dec 02 2009
  • Mathematica
    Table[Count[IntegerPartitions[n],?(Total[1/#]<1&)],{n,60}] (* _Harvey P. Dale, Dec 14 2012 *)

Extensions

Extended by Emeric Deutsch, Dec 02 2009

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

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Jul 16 2018

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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

Programs

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

Extensions

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

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[#]}]]]&]
Previous Showing 31-40 of 43 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.