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.

Previous Showing 51-56 of 56 results.

A356067 Number of integer partitions of n into relatively prime prime-powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 11, 7, 18, 16, 26, 27, 43, 41, 65, 65, 92, 100, 137, 142, 194, 210, 270, 295, 379, 410, 519, 571, 699, 782, 947, 1046, 1267, 1414, 1673, 1870, 2213, 2465, 2897, 3230, 3757, 4210, 4871, 5427, 6265, 6997
Offset: 0

Views

Author

Gus Wiseman, Jul 28 2022

Keywords

Examples

			The a(5) = 1 through a(12) = 7 partitions:
  (32)  .  (43)   (53)   (54)    (73)    (74)     (75)
           (52)   (332)  (72)    (433)   (83)     (543)
           (322)         (432)   (532)   (92)     (552)
                         (522)   (3322)  (443)    (732)
                         (3222)          (533)    (4332)
                                         (542)    (5322)
                                         (722)    (33222)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Crossrefs

This is the relatively prime case of A023894, facs A000688, w/ 1's A023893.
For strict instead of coprime: A054685, facs A050361, with 1's A106244.
The version for factorizations instead of partitions is A354911.
A000041 counts partitions, strict A000009.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A279784 counts twice-partitions where the latter partitions are constant.
A289509 lists numbers whose prime indices are relatively prime.
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A001970, A055887, A063834, A076610, A085970, A355737, A355742.

Programs

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

A357859 Number of integer factorizations of 2n into distinct even factors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 4, 1, 2, 1, 5, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 6, 1, 2, 1, 5, 1, 3, 1, 3, 1, 3, 1, 7, 1, 2, 1, 3, 1, 3, 1, 7, 1, 2, 1, 6, 1, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Oct 17 2022

Keywords

Examples

			The a(n) factorizations for n = 2, 4, 12, 24, 32, 48, 60, 96:
  (4)  (8)    (24)    (48)     (64)     (96)      (120)     (192)
       (2*4)  (4*6)   (6*8)    (2*32)   (2*48)    (2*60)    (2*96)
              (2*12)  (2*24)   (4*16)   (4*24)    (4*30)    (4*48)
                      (4*12)   (2*4*8)  (6*16)    (6*20)    (6*32)
                      (2*4*6)           (8*12)    (10*12)   (8*24)
                                        (2*6*8)   (2*6*10)  (12*16)
                                        (2*4*12)            (4*6*8)
                                                            (2*4*24)
                                                            (2*6*16)
                                                            (2*8*12)

Crossrefs

The version for partitions instead of factorizations is A000009.
Positions of 1's are A004280.
The non-strict version is A340785.
Including odd n gives A357860.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime-power divisors.
A050361 counts strict factorizations into prime powers.
Cf. A000688, A000961, A023894, A295935, A318721.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[2*n],UnsameQ@@#&&OddQ[Times@@(#+1)]&]],{n,100}]

A357860 Number of integer factorizations of n into distinct even factors.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 3, 0, 1, 0
Offset: 1

Views

Author

Gus Wiseman, Oct 17 2022

Keywords

Examples

			The factorizations of 36..48 are (empty columns indicated by dots):
  36    .  38  .  40    .  42  .  44    .  46  .  48
  2*18            2*20            2*22            6*8
                  4*10                            2*24
                                                  4*12
                                                  2*4*6

Crossrefs

The non-strict version is A349906.
Same as A357859 with zeros every other term.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime-power divisors.
A050361 counts strict factorizations into prime powers.
Cf. A000009, A000688, A000961, A023894, A295935, A318721, A340785.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@#&&OddQ[Times@@(#+1)]&]],{n,100}]

A369219 Number of partitions of n into prime power parts (not including 1) not greater than sqrt(n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 2, 2, 3, 2, 3, 3, 10, 8, 12, 10, 14, 12, 16, 14, 19, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 612, 656, 699, 748, 795, 849, 901, 960, 1017, 1081, 1144, 1214, 1282, 1358, 1433, 3620, 3845
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 16 2024

Keywords

Crossrefs

Cf. A023894, A097356, A369218.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - Boole[PrimePowerQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 65}]

A321936 Number of integer partitions of n containing no 1's, prime powers, or squarefree numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 2, 0, 3, 1, 1, 0, 7, 0, 2, 0, 5, 0, 5, 0, 7, 1, 3, 0, 12, 0, 4, 2, 10, 1, 8, 0, 14, 2, 6, 0, 22, 1, 10, 3, 20, 1, 15, 0, 26, 5, 12, 2
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

Number of integer partitions of n using elements of A126706.

Examples

			The a(56) = 7 partitions:
  (56)
  (28,28)
  (36,20)
  (44,12)
  (20,18,18)
  (24,20,12)
  (20,12,12,12)

Crossrefs

Cf. A000607, A000961, A002095, A005117, A023893, A023894, A078135, A114374, A126706, A246655, A321346, A321347, A321378, A321665.

Programs

  • Mathematica
    nn=100;
ser=Product[If[n==1||PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A329097 Expansion of Product_{p prime, k>=1} 1 / (1 + x^(p^k)).

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 1, 0, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -2, 3, -4, 3, -4, 5, -5, 6, -6, 7, -8, 9, -9, 11, -12, 13, -16, 15, -17, 20, -22, 23, -26, 29, -30, 35, -38, 40, -45, 50, -52, 58, -65, 69, -75, 82, -89, 96, -107, 114, -123, 135, -145, 158, -170, 185, -200, 216, -232, 251
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

Comments

Convolution inverse of A054685.

Crossrefs

Cf. A023894, A048165, A054685, A246655, A328556.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1/(1 + Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) Boole[PrimePowerQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]

Formula

G.f.: Product_{k>=1} 1 / (1 + x^A246655(k)).
Previous Showing 51-56 of 56 results.

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