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 11 results. Next

A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609
Offset: 0

Views

Author

Gus Wiseman, Oct 28 2023

Keywords

Examples

			The a(3) = 1 through a(11) = 13 partitions:
  (3)  .  (5)    (3,3)  (7)      (3,3,2)  (9)        (5,5)      (11)
          (3,2)         (4,3)             (5,4)      (4,3,3)    (6,5)
                        (5,2)             (6,3)      (3,3,2,2)  (7,4)
                        (3,2,2)           (7,2)                 (8,3)
                                          (3,3,3)               (9,2)
                                          (4,3,2)               (4,4,3)
                                          (5,2,2)               (5,4,2)
                                          (3,2,2,2)             (6,3,2)
                                                                (7,2,2)
                                                                (3,3,3,2)
                                                                (4,3,2,2)
                                                                (5,2,2,2)
                                                                (3,2,2,2,2)

Crossrefs

This is the odd case of A018783, complement A000837.
The even version is A047967.
The complement is counted by A366850, ranks A366846.
A000041 counts integer partitions, strict A000009.
A000740 counts relatively prime compositions.
A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A289508 gives gcd of prime indices, positions of ones A289509.
Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

A366844 Number of strict integer partitions of n into odd relatively prime parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 5, 4, 4, 5, 6, 7, 8, 8, 9, 11, 12, 12, 15, 16, 15, 19, 23, 23, 26, 28, 30, 34, 37, 38, 44, 48, 48, 56, 62, 63, 72, 77, 82, 92, 96, 102, 116, 124, 128, 142, 155, 162, 178, 191, 200, 222, 236, 246, 276, 291, 303, 334
Offset: 0

Views

Author

Gus Wiseman, Oct 29 2023

Keywords

Examples

			The a(n) partitions for n = 1, 8, 14, 17, 16, 20, 21:
  (1)  (5,3)  (9,5)   (9,5,3)   (9,7)      (11,9)      (9,7,5)
       (7,1)  (11,3)  (9,7,1)   (11,5)     (13,7)      (11,7,3)
              (13,1)  (11,5,1)  (13,3)     (17,3)      (11,9,1)
                      (13,3,1)  (15,1)     (19,1)      (13,5,3)
                                (7,5,3,1)  (9,7,3,1)   (13,7,1)
                                           (11,5,3,1)  (15,5,1)
                                                       (17,3,1)

Crossrefs

This is the relatively prime case of A000700.
The pairwise coprime version is the odd-part case of A007360.
Allowing even parts gives A078374.
The halved even version is A078374 aerated.
The non-strict version is A366843, with evens A000837.
The complement is counted by the strict case of A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A116598, A239261, A302697, A337485, A365067, A366845, A366848.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#==1&]],{n,0,30}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366844(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

Extensions

More terms from 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&]

A366850 Number of integer partitions of n whose odd parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 32, 43, 60, 80, 110, 140, 194, 244, 327, 410, 544, 670, 883, 1081, 1401, 1708, 2195, 2651, 3382, 4069, 5129, 6157, 7708, 9194, 11438, 13599, 16788, 19911, 24432, 28858, 35229, 41507, 50359, 59201, 71489, 83776, 100731, 117784
Offset: 0

Views

Author

Gus Wiseman, Oct 28 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (53)
             (111)  (211)   (221)    (321)     (331)      (71)
                    (1111)  (311)    (411)     (421)      (431)
                            (2111)   (2211)    (511)      (521)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

For all parts (not just odd) we have A000837, complement A018783.
The complement is counted by A366842.
These partitions have ranks A366846.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A007359, A047967, A051424, A066208, A116598, A365067, A366843, A366844, A366845, A366848.

Programs

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

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&]

A366849 Odd numbers whose halved even prime indices are relatively prime.

Original entry on oeis.org

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 91, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 203, 207, 213, 219, 225, 231, 237, 243, 247, 249, 255, 261, 267, 273, 279, 285, 291, 297, 301, 303, 309
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 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}
   9: {2,2}
  15: {2,3}
  21: {2,4}
  27: {2,2,2}
  33: {2,5}
  39: {2,6}
  45: {2,2,3}
  51: {2,7}
  57: {2,8}
  63: {2,2,4}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  87: {2,10}
  91: {4,6}
  93: {2,11}
  99: {2,2,5}

Crossrefs

For odd instead of halved even prime indices we have A366848.
A version for odd indices A366846, counted by A366850.
This is the odd restriction of A366847, counted by A366845.
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.
A289509 lists numbers with relatively prime prime indices, ones of A289508, counted by A000837.
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, A302697, A325698, A366842, A366843.

Programs

  • Mathematica
    Select[Range[100], OddQ[#]&&GCD@@Select[PrimePi/@First/@FactorInteger[#], EvenQ]==2&]

A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 4, 0, 1, 4, 1, 2, 6, 1, 1, 6, 3, 1, 8, 2, 1, 13, 1, 0, 13, 1, 7, 15, 1, 1, 19, 6, 1, 25, 1, 2, 33, 1, 1, 32, 5, 10, 39, 2, 1, 46, 14, 6, 55, 1, 1, 77, 1, 1, 82, 0, 20, 92, 1, 2, 105, 31, 1, 122, 1, 1, 166, 2, 16, 168
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Examples

			The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
(3)  (9)      (15)         (21)             (25)         (27)
     (3,3,3)  (5,5,5)      (7,7,7)          (15,5,5)     (9,9,9)
              (9,3,3)      (9,9,3)          (5,5,5,5,5)  (15,9,3)
              (3,3,3,3,3)  (15,3,3)                      (21,3,3)
                           (9,3,3,3,3)                   (9,9,3,3,3)
                           (3,3,3,3,3,3,3)               (15,3,3,3,3)
                                                         (9,3,3,3,3,3,3)
                                                         (3,3,3,3,3,3,3,3,3)

Links

Crossrefs

Allowing even parts gives A018783, complement A000837.
For parts > 1 instead of gcd > 1 we have A087897.
For gcd = 1 instead of gcd > 1 we have A366843.
The strict case is A366750, with evens A303280.
The strict complement is A366844, with evens A078374.
A000041 counts integer partitions, strict A000009 (also into odd parts).
A000700 counts strict partitions into odd parts.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366852(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

Extensions

More terms from Chai Wah Wu, Nov 02 2023
a(0)=0 prepended by Alois P. Heinz, Jan 11 2024

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&]

A366853 Number of integer partitions of n into odd, pairwise coprime parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 18, 20, 22, 25, 29, 33, 36, 39, 43, 49, 55, 61, 66, 69, 75, 85, 94, 104, 113, 120, 129, 143, 159, 172, 183, 193, 207, 226, 251, 272, 288, 304, 325, 350, 383, 414, 437, 460, 494, 532, 577, 622, 655, 684
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Examples

			The a(1) = 1 through a(10) = 7 partitions:
1  11  3    31    5      51      7        53        9          73
       111  1111  311    3111    511      71        531        91
                  11111  111111  31111    5111      711        5311
                                 1111111  311111    51111      7111
                                          11111111  3111111    511111
                                                    111111111  31111111
                                                               1111111111

Crossrefs

Partitions into odd parts are counted by A000009, ranks A066208.
Allowing even parts gives A051424.
For relatively prime (not pairwise coprime): A366843, with evens A000837.
A000041 counts integer partitions, strict A000009 (also into odds).
A101268 counts pairwise coprime compositions.
A168532 counts partitions by gcd.
Cf. A007359, A018783, A055922, A078374, A337485, A366844, A366852.

Programs

  • Mathematica
    pwcop[y_]:=And@@(GCD@@#==1&)/@Subsets[y,{2}]
Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&pwcop[#]&]],{n,0,30}]
Showing 1-10 of 11 results. Next

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