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 1-5 of 5 results.

A319161 Numbers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

Numbers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (211), (6), (1111), (7), (221), (8), (311), (9), (2111), (33), (222), (411).

Crossrefs

Cf. A001597, A007916, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A305732, A319151, A319160, A319163, A319165.

Programs

  • Mathematica
    Select[Range[100],GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1&]

A319164 Number of integer partitions of n that are neither relatively prime nor aperiodic.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 5, 0, 2, 2, 5, 0, 6, 0, 9, 2, 2, 0, 17, 1, 2, 3, 17, 0, 18, 0, 22, 2, 2, 2, 48, 0, 2, 2, 48, 0, 34, 0, 58, 11, 2, 0, 111, 1, 14, 2, 103, 0, 65, 2, 141, 2, 2, 0, 264, 0, 2, 19, 231, 2, 116, 0, 299, 2, 42
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A partition is aperiodic if its multiplicities are relatively prime.

Examples

			The a(24) = 17 integer partitions:
  (12,12),
  (8,8,8),
  (6,6,6,6), (8,8,4,4), (9,9,3,3), (10,10,2,2),
  (4,4,4,4,4,4), (6,6,3,3,3,3), (6,6,4,4,2,2), (6,6,6,2,2,2), (8,8,2,2,2,2),
  (3,3,3,3,3,3,3,3), (4,4,4,4,2,2,2,2), (6,6,2,2,2,2,2,2),
  (4,4,4,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2,2,2).

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319165.

Programs

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

A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 144, 169, 243, 256, 289, 324, 343, 361, 400, 512, 529, 576, 625, 729, 784, 841, 961, 1024, 1331, 1369, 1600, 1681, 1728, 1849, 1936, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2704, 2809, 2916, 3125
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

Perfect powers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (22), (1111), (33), (222), (11111), (44), (111111), (2222), (55), (333), (1111111), (221111), (66), (22222), (11111111), (77), (222211), (444), (88), (331111), (111111111), (99), (22111111), (3333), (222222), (441111).

Crossrefs

Cf. A001597, A007916, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319162, A319165.

Programs

  • Mathematica
    Select[Range[1000],And[GCD@@FactorInteger[#][[All,2]]>1,GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1]&]

A319180 Perfect powers whose prime indices are relatively prime.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 100, 128, 144, 196, 216, 225, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1089, 1156, 1225, 1296, 1444, 1600, 1728, 1764, 1936, 2025, 2048, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3364, 3375, 3600, 3844, 4096
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (3311), (1111111), (221111), (4411), (222111), (3322), (11111111), (222211), (331111), (5511), (111111111), (22111111), (6611), (441111), (332211), (333111).

Crossrefs

Cf. A001597, A056239, A072774, A289509, A298748, A319161, A319163, A319165, A319179, A319181.

Programs

  • Mathematica
    Select[Range[1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,GCD@@FactorInteger[#][[All,2]]>1]&]

A319181 Numbers that are not perfect powers but whose prime indices have a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 171, 173, 179, 181, 183, 185, 189
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (6,2), (13), (14), (15), (16), (8,2), (17), (18), (4,2,2), (6,3).

Crossrefs

Cf. A007916, A056239, A072774, A289509, A298748, A319161, A319163, A319165, A319179, A319180.

Programs

  • Mathematica
    Select[Range[1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]>1,GCD@@FactorInteger[#][[All,2]]==1]&]
Showing 1-5 of 5 results.

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