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 1-4 of 4 results.

A047968 a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.

Original entry on oeis.org

1, 3, 4, 8, 8, 17, 16, 30, 34, 52, 57, 99, 102, 153, 187, 261, 298, 432, 491, 684, 811, 1061, 1256, 1696, 1966, 2540, 3044, 3876, 4566, 5846, 6843, 8610, 10203, 12610, 14906, 18491, 21638, 26508, 31290, 38044, 44584, 54133, 63262, 76241
Offset: 1

Views

Author

N. J. A. Sloane, Dec 11 1999

Keywords

Comments

Inverse Moebius transform of A000041.
Row sums of triangle A137587. - Gary W. Adamson, Jan 27 2008
Row sums of triangle A168021. - Omar E. Pol, Nov 20 2009
Row sums of triangle A168017. Row sums of triangle A168018. - Omar E. Pol, Nov 25 2009
Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014
Conjecture: for n > 6, a(n) is strictly increasing. - Franklin T. Adams-Watters, Apr 19 2014
Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018

Examples

			For n = 10 the divisors of 10 are 1, 2, 5, 10, hence the partition numbers of the divisors of 10 are 1, 2, 7, 42, so a(10) = 1 + 2 + 7 + 42 = 52. - _Omar E. Pol_, Feb 26 2014
From _Gus Wiseman_, Sep 16 2018: (Start)
The a(6) = 17 constant multiset partitions:
  (111111)  (111)(111)    (11)(11)(11)  (1)(1)(1)(1)(1)(1)
  (111222)  (12)(12)(12)
  (111122)  (112)(112)
  (112233)  (123)(123)
  (111112)
  (111123)
  (111223)
  (111234)
  (112234)
  (112345)
  (123456)
(End)

Links

Crossrefs

Cf. A000041, A000837, A047966, A055893, A137587, A003606 (Euler transform).
Cf. A002033, A003238, A018783, A034729, A052409, A078392, A100953, A319162.

Programs

  • Maple
    with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l) do c := c+numbpart(l[i]) od: RETURN(c): end: for j from 1 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007
  • Mathematica
    a[n_] := Sum[ PartitionsP[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Oct 03 2013 *)

Formula

G.f.: Sum_{k>0} (-1+1/Product_{i>0} (1-z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: sum(n>0,A000041(n)*x^n/(1-x^n)). - Mircea Merca, Feb 24 2014.
a(n) = A168111(n) + A000041(n). - Omar E. Pol, Feb 26 2014
a(n) = Sum_{y is a partition of n} A000005(GCD(y)). - Gus Wiseman, Sep 16 2018

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 3, 9, 9, 12, 12, 20, 18, 28, 27, 37, 42, 55, 51, 74, 80, 98, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144
Offset: 0

Views

Author

Gus Wiseman, Apr 15 2021

Keywords

Comments

Alternative name: Number of strict integer partitions of n with no part dividing all the others.

Examples

			The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
  0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
                         52        72    73    74    543   85    95     96
                                   432   532   83    732   94    A4     B4
                                               92          A3    B3     D2
                                               542         B2    653    654
                                               632         643   743    753
                                                           652   752    762
                                                           742   932    843
                                                           832   5432   852
                                                                        942
                                                                        A32
                                                                        6432

Crossrefs

The complement is counted by A097986 (non-strict: A083710, rank: A339563).
The complement with no 1's is A098965 (non-strict: A083711).
The non-strict version is A338470.
The Heinz numbers of these partitions are A339562 (non-strict: A342193).
The case with greatest part not divisible by all others is A343379.
The case with greatest part divisible by all others is A343380.
A000009 counts strict partitions (non-strict: A000041).
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]

Formula

a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483
Offset: 0

Views

Author

Gus Wiseman, Mar 23 2021

Keywords

Comments

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

Examples

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Links

Crossrefs

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)
  • PARI
    a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

Formula

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 0, 2, 5, 1, 0, 0, 0, 7, 1, 0, 0, 2, 3, 11, 1, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 2, 0, 5, 22, 1, 0, 0, 0, 0, 0, 3, 0, 30, 1, 0, 0, 0, 0, 2, 0, 0, 7, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101
Offset: 1

Views

Author

Omar E. Pol, Nov 21 2009

Keywords

Examples

			Triangle begins:
==============================================
.... k: 12 11 10. 9. 8. 7. 6. 5. 4. 3.. 2.. 1.
==============================================
n=1 ....................................... 1,
n=2 ................................... 1,  2,
n=3 ............................... 1,  0,  3,
n=4 ............................ 1, 0,  2,  5,
n=5 ......................... 1, 0, 0,  0,  7,
n=6 ...................... 1, 0, 0, 2,  3, 11,
n=7 ................... 1, 0, 0, 0, 0,  0, 15,
n=8 ................ 1, 0, 0, 0, 2, 0,  5, 22,
n=9 ............. 1, 0, 0, 0, 0, 0, 3,  0, 30,
n=10 ......... 1, 0, 0, 0, 0, 2, 0, 0,  7, 42,
n=11 ...... 1, 0, 0, 0, 0, 0, 0, 0, 0,  0, 56,
n=12 ... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77,
...

Links

Crossrefs

Cf. A000005, A000007, A000041, A035363, A035444, A047968, A135010.
Cf. A138121, A168014, A168015, A168020, A168021, A168111.

Programs

  • Mathematica
    T[n_, k_]:= If[IntegerQ[n/(n-k+1)], PartitionsP[n/(n-k+1)], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)
  • SageMath
    def T(n,k): return number_of_partitions(n/(n-k+1)) if (n%(n-k+1))==0 else 0
flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

Formula

T(n, k) = A000041(n/k) if k|n; otherwise T(n,k) = 0.
T(n, n) = A000041(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(2*n, n) = A000007(n-1).
Sum_{k=1..n} T(n, k) = A047968(n).
Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)

Extensions

Edited and extended by Max Alekseyev, May 07 2010
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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