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 1-6 of 6 results.

A358901 Number of integer partitions of n whose parts have all different numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 7, 9, 8, 9, 11, 11, 15, 16, 16, 18, 20, 22, 26, 28, 31, 32, 36, 40, 45, 46, 46, 50, 59, 64, 70, 75, 78, 83, 89, 94, 108, 106, 104, 120, 137, 142, 147, 150, 161, 174, 190, 200, 220, 226, 224, 248, 274, 274, 287, 301, 320, 340, 351, 361
Offset: 0

Views

Author

Gus Wiseman, Dec 07 2022

Keywords

Examples

			The a(1) = 1 through a(11) = 7 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (41)  (42)  (43)   (62)   (54)   (82)   (74)
                              (51)  (61)   (71)   (63)   (91)   (65)
                                    (421)  (431)  (81)   (451)  (83)
                                                  (621)  (631)  (92)
                                                                (A1)
                                                                (821)

Links

Crossrefs

The weakly decreasing version is A358909 (complement A358910).
The version not counting multiplicity is A358903, weakly decreasing A358902.
For equal numbers of prime factors we have A319169, compositions A358911.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A056239, A129519, A141199, A218482, A300335, A319071, A320324, A358335, A358831, A358908.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeOmega/@#&]],{n,0,60}]

Extensions

a(61) and beyond from Lucas A. Brown, Dec 14 2022

A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 7, 9, 12, 20, 21, 39, 49, 79, 109, 161, 236, 345, 512, 752, 1092, 1628, 2376, 3537, 5171, 7650, 11266, 16634, 24537, 36173, 53377, 78791, 116224, 171598, 253109, 373715, 551434, 814066, 1201466, 1773425, 2617744, 3864050, 5703840, 8419699
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2022

Keywords

Examples

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (23)     (33)      (25)       (35)
                    (1111)  (32)     (222)     (52)       (44)
                            (11111)  (111111)  (223)      (53)
                                               (232)      (233)
                                               (322)      (323)
                                               (1111111)  (332)
                                                          (2222)
                                                          (11111111)

Links

Crossrefs

The case of partitions is A319169, ranked by A320324.
The weakly decreasing version is A358335, strictly A358901.
For sequences of partitions see A358905.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A358902 = compositions with weakly decreasing A001221, strictly A358903.
A358909 = partitions with weakly decreasing A001222, complement A358910.
Cf. A056239, A063834, A064573, A218482, A279787, A300335, A319066, A319071, A358831, A358908.

Programs

  • Maple
    b:= proc(n, i) option remember; uses numtheory; `if`(n=0, 1, add(
     (t-> `if`(i<0 or i=t, b(n-j, t), 0))(bigomega(j)), j=1..n))
    end:
a:= n-> b(n, -1):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 12 2024
  • Mathematica
    Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,0,10}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 68, 100, 153, 227, 342, 509, 759, 1129, 1678, 2492, 3699, 5477, 8121, 12015, 17795, 26313, 38924, 57541, 85065, 125712, 185758, 274431, 405420, 598815, 884465, 1306165, 1928943, 2848360, 4205979, 6210289, 9169540
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2022

Keywords

Examples

			The a(0) = 1 through a(6) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (33)
                 (111)  (31)    (32)     (42)
                        (211)   (41)     (51)
                        (1111)  (221)    (222)
                                (311)    (231)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Links

Crossrefs

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358901.
The version not counting multiplicity is A358902, strict A358903.
The case of partitions is A358909, complement A358910.
The case of equality is A358911, partitions A319169.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.

Programs

  • Mathematica
    Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,10}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 84, 134, 213, 338, 536, 850, 1349, 2136, 3389, 5367, 8509, 13480, 21362, 33843, 53624, 84957, 134600, 213251, 337850, 535251, 847987, 1343440, 2128372, 3371895, 5341977, 8463051, 13407689, 21241181, 33651507, 53312538, 84460690
Offset: 0

Views

Author

Gus Wiseman, Dec 07 2022

Keywords

Examples

			The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (24)
                 (111)  (31)    (32)     (33)
                        (211)   (41)     (42)
                        (1111)  (221)    (51)
                                (311)    (222)
                                (2111)   (231)
                                (11111)  (321)
                                         (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Links

Crossrefs

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358903.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A116608 counts partitions by sum and number of distinct parts.
A334028 counts distinct parts in standard compositions.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A056239, A071625, A129519, A300335, A358831, A358908, A358911.

Programs

  • Maple
    p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t<=i, b(n-j, t), 0))(p(j)), j=1..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 14 2024
  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeNu/@#&]],{n,0,10}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306
Offset: 0

Views

Author

Gus Wiseman, Dec 09 2022

Keywords

Comments

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

Crossrefs

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605
Offset: 0

Views

Author

Gus Wiseman, Dec 09 2022

Keywords

Examples

			The a(9) = 1 through a(14) = 11 partitions:
  (54)  (541)  (74)    (543)    (76)      (554)
               (542)   (741)    (544)     (743)
               (5411)  (5421)   (742)     (761)
                       (54111)  (5422)    (5432)
                                (5431)    (5441)
                                (7411)    (7421)
                                (54211)   (54221)
                                (541111)  (54311)
                                          (74111)
                                          (542111)
                                          (5411111)

Crossrefs

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The complement is counted by A358909.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358902, A358903, A358908.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]
Showing 1-6 of 6 results.

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