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.

A366843 Number of integer partitions of n into odd, relatively prime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 13, 17, 21, 23, 32, 37, 42, 53, 62, 70, 88, 103, 116, 139, 164, 184, 220, 255, 283, 339, 390, 435, 511, 578, 653, 759, 863, 963, 1107, 1259, 1401, 1609, 1814, 2015, 2303, 2589, 2878, 3259, 3648, 4058, 4580, 5119, 5672, 6364
Offset: 0

Views

Author

Gus Wiseman, Oct 28 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (31)    (311)    (51)      (331)      (53)
                    (1111)  (11111)  (3111)    (511)      (71)
                                     (111111)  (31111)    (3311)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Crossrefs

Allowing even parts gives A000837.
The strict case is A366844, with evens A078374.
The complement is counted by A366852, with evens A018783.
The pairwise coprime version is A366853, with evens A051424.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A113685, A116598, A289509, A289508, A302697, A337485, A366845, A366848, A366849.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],#=={}||And@@OddQ/@#&&GCD@@#==1&]],{n,0,30}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366843(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

A366845 Number of integer partitions of n that contain at least one even part and whose halved even parts are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 43, 58, 82, 107, 144, 189, 250, 323, 420, 537, 695, 880, 1114, 1404, 1774, 2210, 2759, 3423, 4239, 5223, 6430, 7869, 9640, 11738, 14266, 17297, 20950, 25256, 30423, 36545, 43824, 52421, 62620, 74599, 88802, 105431
Offset: 0

Views

Author

Gus Wiseman, Oct 28 2023

Keywords

Examples

			The partition y = (6,4) has halved even parts (3,2) which are relatively prime, so y is counted under a(10).
The a(2) = 1 through a(9) = 15 partitions:
  (2)  (21)  (22)   (32)    (42)     (52)      (62)       (72)
             (211)  (221)   (222)    (322)     (332)      (432)
                    (2111)  (321)    (421)     (422)      (522)
                            (2211)   (2221)    (521)      (621)
                            (21111)  (3211)    (2222)     (3222)
                                     (22111)   (3221)     (3321)
                                     (211111)  (4211)     (4221)
                                               (22211)    (5211)
                                               (32111)    (22221)
                                               (221111)   (32211)
                                               (2111111)  (42111)
                                                          (222111)
                                                          (321111)
                                                          (2211111)
                                                          (21111111)

Crossrefs

For all parts we have A000837, complement A018783.
These partitions have ranks A366847.
For odd parts we have A366850, ranks A366846, complement A366842.
A000041 counts integer partitions, strict A000009, complement A047967.
A035363 counts partitions into all even parts, ranks A066207.
A078374 counts relatively prime strict partitions.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A051424, A113685, A116598, A258117, A289509, A366843, A366849.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,EvenQ]/2==1&]],{n,0,30}]

A366848 Odd numbers whose odd prime indices are relatively prime.

Original entry on oeis.org

55, 85, 155, 165, 187, 205, 253, 255, 275, 295, 335, 341, 385, 391, 415, 425, 451, 465, 485, 495, 527, 545, 561, 595, 605, 615, 635, 649, 697, 713, 715, 737, 745, 759, 765, 775, 785, 799, 803, 825, 885, 895, 913, 935, 943, 955, 1003, 1005, 1023, 1025, 1045
Offset: 1

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The odd prime indices of 345 are {3,9}, which are not relatively prime, so 345 is not in the sequence.
The odd prime indices of 825 are {3,3,5}, which are relatively prime, so 825 is in the sequence
The terms together with their prime indices begin:
    55: {3,5}
    85: {3,7}
   155: {3,11}
   165: {2,3,5}
   187: {5,7}
   205: {3,13}
   253: {5,9}
   255: {2,3,7}
   275: {3,3,5}
   295: {3,17}
   335: {3,19}
   341: {5,11}
   385: {3,4,5}
   391: {7,9}
   415: {3,23}
   425: {3,3,7}
   451: {5,13}
   465: {2,3,11}
   485: {3,25}
   495: {2,2,3,5}

Crossrefs

Including even terms and prime indices gives A289509, ones of A289508, counted by A000837.
Including even prime indices gives A302697, counted by A302698.
Including even terms gives A366846, counted by A366850.
For halved even instead of odd prime indices we have A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A066208 lists numbers with all odd prime indices, even A066207.
A112798 lists prime indices, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A018783, A055396, A061395, A087436, A325698, A365067, A366842, A366843, A366844, A366847.

Programs

  • Mathematica
    Select[Range[1000], OddQ[#]&&GCD@@Select[PrimePi/@First/@FactorInteger[#], OddQ]==1&]

A366846 Numbers whose odd prime indices are relatively prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122
Offset: 1

Views

Author

Gus Wiseman, Oct 29 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The odd prime indices of 115 are {3,9}, and these are not relatively prime, so 115 is not in the sequence.
The odd prime indices of 825 are {3,3,5}, and these are relatively prime, so 825 is in the sequence.

Crossrefs

Including even indices gives A289509, ones of A289508, counted by A000837.
The complement when including even indices is A318978, counted by A018783.
The nonzero complement ranks the partitions counted by A366842.
The version for halved even indices is A366847.
The odd case is A366848.
The partitions with these Heinz numbers are counted by A366850.
A000041 counts integer partitions, strict A000009 (also into odds).
A112798 lists prime indices, length A001222, sum A056239.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A087436, A302696, A302697, A325698, A366843, A366844, A366849.

Programs

  • Mathematica
    Select[Range[100], GCD@@Select[PrimePi/@First/@FactorInteger[#], OddQ]==1&]

A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174
Offset: 1

Views

Author

Gus Wiseman, Oct 31 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Consists of powers of 2 times elements of the odd restriction A366849.

Examples

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Crossrefs

Including odd indices gives A289509, ones of A289508, counted by A000837.
The complement including odd indices is A318978, counted by A018783.
The partitions with these ranks are counted by A366845.
A version for odd indices A366846, counted by A366850.
The odd restriction is A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A168532, A302696, A302697, A325698, A366842, A366843, A366844, A366848.

Programs

  • Mathematica
    Select[Range[100],GCD@@Select[PrimePi/@First/@FactorInteger[#],EvenQ]/2==1&]
Showing 1-5 of 5 results.

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