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 1-7 of 7 results.

A320322 Number of integer partitions of n whose product is a perfect power.

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 5, 5, 9, 11, 18, 19, 28, 30, 42, 50, 68, 76, 102, 113, 146, 170, 212, 241, 312, 356, 441, 514, 628, 720, 887, 1008, 1215, 1403, 1660, 1903, 2291, 2609, 3107, 3594, 4254, 4864, 5739, 6546, 7672, 8811, 10237, 11651, 13583, 15420, 17867, 20382
Offset: 0

Views

Author

Gus Wiseman, Oct 10 2018

Keywords

Examples

			The a(4) = 2 through a(11) = 19 integer partitions:
  4   41   33    331    8       9        55        551
  22  221  42    421    44      81       82        632
           222   2221   422     333      91        821
           411   4111   2222    441      433       911
           2211  22111  3311    4221     442       4331
                        4211    22221    811       4421
                        22211   33111    3322      8111
                        41111   42111    3331      33221
                        221111  222111   4222      33311
                                411111   4411      42221
                                2211111  22222     44111
                                         42211     222221
                                         222211    422111
                                         331111    2222111
                                         421111    3311111
                                         2221111   4211111
                                         4111111   22211111
                                         22111111  41111111
                                                   221111111

Crossrefs

Cf. A003963, A064573, A305551, A319056, A319066, A319071, A319169, A320325.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],GCD@@FactorInteger[Times@@#][[All,2]]>1&]],{n,30}]

A064573 Number of partitions of n into parts which are all powers of the same prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 13, 15, 20, 21, 29, 30, 37, 40, 50, 51, 64, 65, 80, 84, 99, 100, 123, 125, 146, 151, 178, 179, 212, 213, 249, 255, 292, 295, 348, 349, 396, 404, 466, 467, 535, 536, 611, 622, 697, 698, 801, 803, 900, 910, 1025, 1026, 1152, 1156, 1298, 1311
Offset: 1

Views

Author

Marc LeBrun, Sep 20 2001

Keywords

Comments

The exponents cannot all be zero.

Examples

			a(5)=5: 5^1, 3^1+2*3^0, 2^2+1, 2*2^1+1, 2^1+3*2^0
From _Gus Wiseman_, Oct 10 2018: (Start)
The a(2) = 1 through a(9) = 15 integer partitions:
  (2)  (3)   (4)    (5)     (33)     (7)       (8)        (9)
       (21)  (22)   (41)    (42)     (331)     (44)       (81)
             (31)   (221)   (51)     (421)     (71)       (333)
             (211)  (311)   (222)    (511)     (422)      (441)
                    (2111)  (411)    (2221)    (2222)     (711)
                            (2211)   (4111)    (3311)     (4221)
                            (3111)   (22111)   (4211)     (22221)
                            (21111)  (31111)   (5111)     (33111)
                                     (211111)  (22211)    (42111)
                                               (41111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (411111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)
(End)

Links

Crossrefs

Cf. A028422, A064572, A064574, A064575, A064576, A064577, A319071, A320322, A320325.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],PrimePowerQ[Times@@#]&]],{n,30}] (* Gus Wiseman, Oct 10 2018 *)
  • PARI
    first(n)={Vec(sum(k=2, n, if(isprime(k), 1/prod(r=0, logint(n,k), 1-x^(k^r) + O(x*x^n)) - 1/(1-x), 0)), -n)} \\ Andrew Howroyd, Dec 29 2017

Formula

G.f.: Sum_{k>=1} 1/(Product_{r>=0} 1-x^(prime(k)^r)) - 1/(1-x). - Andrew Howroyd, Dec 29 2017

Extensions

Name clarified by Andrew Howroyd, Dec 29 2017

A319169 Number of integer partitions 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, 3, 4, 4, 6, 6, 8, 7, 11, 11, 14, 15, 20, 19, 26, 27, 34, 35, 43, 45, 59, 60, 72, 77, 94, 98, 118, 125, 148, 158, 184, 198, 233, 245, 282, 308, 353, 374, 428, 464, 525, 566, 635, 686, 779, 832, 930, 1005, 1123, 1208, 1345, 1451, 1609, 1732, 1912
Offset: 0

Views

Author

Gus Wiseman, Oct 10 2018

Keywords

Examples

			The a(1) = 1 through a(9) = 6 integer partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    32     33      52       44        72
              1111  11111  222     322      53        333
                           111111  1111111  332       522
                                            2222      3222
                                            11111111  111111111

Links

Crossrefs

Cf. A000607, A001222, A003963, A064573, A279787, A305551, A319056, A319066, A319071, A320322, A320324.

Programs

  • Maple
    b:= proc(n, i, f) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, f)+(o-> `if`(f in {0, o}, b(n-i, min(i, n-i),
     `if`(f=0, o, f)), 0))(numtheory[bigomega](i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..75);  # Alois P. Heinz, Dec 15 2018
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,30}]
(* Second program: *)
b[n_, i_, f_] := b[n, i, f] = If[n == 0, 1, If[i < 1, 0,
     b[n, i-1, f] + Function[o, If[f == 0 || f == o, b[n-i, Min[i, n-i],
     If[f == 0, o, f]], 0]][PrimeOmega[i]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 75] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

Extensions

a(51)-a(58) from Chai Wah Wu, Nov 12 2018

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

A320323 Numbers whose product of prime indices (A003963) is a perfect power and where each prime index has the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

7, 9, 19, 23, 25, 27, 49, 53, 81, 97, 103, 121, 125, 131, 151, 161, 169, 225, 227, 243, 289, 311, 343, 361, 419, 529, 541, 625, 661, 679, 691, 719, 729, 827, 841, 961, 1009, 1089, 1127, 1159, 1183, 1193, 1321, 1331, 1369, 1427, 1543, 1589, 1619, 1681, 1849
Offset: 1

Views

Author

Gus Wiseman, Oct 10 2018

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 terms together with their corresponding multiset multisystems (A302242):
    7: {{1,1}}
    9: {{1},{1}}
   19: {{1,1,1}}
   23: {{2,2}}
   25: {{2},{2}}
   27: {{1},{1},{1}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   81: {{1},{1},{1},{1}}
   97: {{3,3}}
  103: {{2,2,2}}
  121: {{3},{3}}
  125: {{2},{2},{2}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}

Crossrefs

Cf. A000720, A001222, A003963, A056239, A064573, A112798, A302242, A305551, A306017, A319056, A319066, A319071, A320324, A320325.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[GCD@@FactorInteger[Times@@primeMS[#]][[All,2]]>1,SameQ@@PrimeOmega/@primeMS[#]]&]
  • PARI
    is(n) = my (f=factor(n), pi=apply(primepi, f[,1]~)); #Set(apply(bigomega, pi))==1 && ispower(prod(i=1, #pi, pi[i]^f[i,2])) \\ Rémy Sigrist, Oct 11 2018
Showing 1-7 of 7 results.

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