A144300 -id:A144300 - cp's OEIS frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

1, 4, 6, 12, 10, 24, 14, 32, 27, 40, 22, 72, 26, 56, 60, 80, 34, 108, 38, 120, 84, 88, 46, 192, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 324, 74, 152, 156, 320, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 448, 228, 232, 118

Offset: 1

Christian G. Bower

Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - Michael Somos, Jun 08 2000
Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008
Sum of all parts of all partitions of n into equal parts. - Omar E. Pol, Jan 18 2013

			For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The sum of all parts is 6 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 24 equalling 6 times the number of divisors of 6, so a(6) = 24. - _Omar E. Pol_, May 08 2021

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

T. D. Noe, Table of n, a(n) for n = 1..1000
J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.
Jean Bourgain, Sergei Konyagin and Igor Shparlinski. Distribution on elements of cosets of small subgroups and applications, arXiv:1103.0567 [math.NT], Mar 2 2011.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147. [Broken link?]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147.

Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.
Cf. A127648, A127093, A127528.
Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).
Column 1 of A329323.

a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory): A038040 := n->tau(n)*n;

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

n*numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])

a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(x^k-1)^2,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */

a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020

from sympy import divisor_count as d
def a(n): return n*d(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 15 2022

[n*sigma(n,0) for n in range(1, 60)] # Stefano Spezia, Jul 20 2025

Dirichlet g.f.: zeta(s-1)^2.
G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - Vladeta Jovovic, Dec 30 2001
Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - Vladeta Jovovic, Oct 30 2001
Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007
Equals row sums of triangle A127528. - Gary W. Adamson, May 21 2007
a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - Omar E. Pol, Jan 18 2013
a(n) = A000203(n) * A240471(n) + A106315(n). - Reinhard Zumkeller, Apr 06 2014
L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017
a(n) = Sum_{d|n} A018804(d). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{d|n} phi(d)*sigma(n/d). - Ridouane Oudra, Jan 21 2021
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 + 2*n*q^n - n^2*q^(2*n))/(1 - q^n)^2. - Peter Bala, Jan 22 2021
a(n) = Sum_{k=1..n} sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/sqrt(log x). That is, there are 0 < A < B such that Ax/sqrt(log x) < f(x) < Bx/sqrt(log x). - Charles R Greathouse IV, Mar 15 2022
Sum_{k=1..n} a(k) ~ n^2*log(n)/2 + (gamma - 1/4)*n^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 25 2022
Mobius transform of A060640. - R. J. Mathar, Feb 07 2023

A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 21, 49, 106, 226, 470, 968, 1971, 3995, 8057, 16208, 32537, 65239, 130687, 261654, 523661, 1047784, 2096150, 4193049, 8387033, 16775258, 33551996, 67105854, 134214010, 268430891, 536865308, 1073734982, 2147475299, 4294957153, 8589922282

Offset: 0

Alford Arnold, Aug 29 2000

Previous name was: Counts members of A056808 by number of factors.
A056808 relates to least prime signatures (cf. A025487)
a(n) is also the number of compositions of n that are not partitions of n. - Omar E. Pol, Jan 31 2009, Oct 14 2013
a(n) is the number of compositions of n into positive parts containing pattern [1,2]. - Bob Selcoe, Jul 08 2014

			A011782 begins     1 1 2 4 8 16 32 64 128 256 ...;
A000041 begins     1 1 2 3 5  7 11 15  22  30 ...;
so sequence begins 0 0 0 1 3  9 21 49 106 226 ... .
For n = 3 the factorizations are 8=2*2*2, 12=2*2*3, 18=2*3*3 and 30=2*3*5.
a(5) = 9: {[1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,3,1], [1,4], [2,1,2], [2,3]}. - _Bob Selcoe_, Jul 08 2014

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A025487, A056808, A117989.
The version for patterns is A002051.
(1,2)-avoiding compositions are just partitions A000041.
The (1,1)-matching version is A261982.
The version for prime indices is A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns matched by compositions are counted by A335456.
Cf. A011782, A056986, A106356, A144300, A238279, A269134, A333755.

a:= n-> ceil(2^(n-1))-combinat[numbpart](n):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 30 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!GreaterEqual@@#&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)
a[n_] := If[n == 0, 0, 2^(n-1) - PartitionsP[n]];
a /@ Range[0, 37] (* Jean-François Alcover, May 23 2021 *)

a(n) = A011782(n) - A000041(n).
a(n) = 2*a(n-1) + A117989(n-1). - Bob Selcoe, Apr 11 2014
G.f.: (1 - x) / (1 - 2*x) - Product_{k>=1} 1 / (1 - x^k). - Ilya Gutkovskiy, Jan 30 2020

More terms from James Sellers, Aug 31 2000

New name from Joerg Arndt, Sep 02 2013

A239955 Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 27, 38, 54, 75, 104, 137, 187, 245, 322, 418, 542, 691, 887, 1121, 1417, 1777, 2228, 2767, 3441, 4247, 5235, 6424, 7871, 9594, 11688, 14173, 17168, 20723, 24979, 30008, 36010, 43085, 51479, 61357, 73032, 86718, 102852, 121718

Offset: 0

Clark Kimberling, Mar 30 2014

From Gus Wiseman, Jun 26 2022: (Start)
Also the number of partitions of n with at least one gap, i.e., partitions whose parts do not form a contiguous interval. These partitions are ranked by A073492. For example, the a(0) = 0 through a(8) = 12 partitions are:
  .  .  .  .  (31)  (41)   (42)    (52)     (53)
                    (311)  (51)    (61)     (62)
                           (411)   (331)    (71)
                           (3111)  (421)    (422)
                                   (511)    (431)
                                   (4111)   (521)
                                   (31111)  (611)
                                            (3311)
                                            (4211)
                                            (5111)
                                            (41111)
                                            (311111)
Also the number of non-constant partitions of n with a repeated non-maximal part, ranked by A065201. The a(0) = 0 through a(8) = 12 partitions are:
  .  .  .  .  (211)  (311)   (411)    (322)     (422)
                     (2111)  (2211)   (511)     (611)
                             (3111)   (3211)    (3221)
                             (21111)  (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)
(End)

			a(6) counts these 4 partitions:  51, 42, 411, 3111.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from John Tyler Rascoe)

Cf. A239954, A239956, A239958.
The complement is counted by A034296 (strict A137793), ranked by A073491.
These partitions are ranked by A073492, conjugate A065201.
Applying the condition to the conjugate gives A350839, ranked by A350841.
A000041 counts integer partitions, strict A000009.
A090858 counts partitions with a single hole, ranked by A325284.
A116931 counts partitions with differences != -1, strict A003114.
A116932 counts partitions with differences != -1 or -2, strict A025157.
Cf. A000070, A001227, A008284, A144300, A183558, A321440.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> combinat[numbpart](n)-add(b(n, k), k=0..n):
seq(a(n), n=0..47);  # Alois P. Heinz, Aug 18 2025

z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}]  (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}]  (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
(* second program *)
Table[Length[Select[IntegerPartitions[n],Min@@Differences[#]<-1&]],{n,0,30}] (* Gus Wiseman, Jun 26 2022 *)

qs(a,q,n) = {prod(k=0,n,1-a*q^k)}
A_q(N) = {if(N<4, vector(N+1,i,0), my(q='q+O('q^(N-2)), g= sum(i=2,N+1, q^i/qs(q,q,i-1)*sum(j=1,i-1, q^(2*j)*qs(q^2,q^2,j-2)))); concat([0,0,0,0], Vec(g)))} \\ John Tyler Rascoe, Aug 16 2025

a(n) = A000041(n) - A034296(n).

G.f.: Sum_{i>1} q^i/(q;q){i-1} * Sum{j=1..i-1} (q^2;q^2){j-2} where (a;q)_k = Product{i>=0..k} (1-a*q^i). - John Tyler Rascoe, Aug 16 2025

A351204 Number of integer partitions of n such that every permutation has all distinct runs.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 9, 11, 14, 18, 20, 25, 28, 34, 41, 47, 53, 64, 72, 84, 98, 113, 128, 148, 169, 194, 223, 255, 289, 333, 377, 428, 488, 554, 629, 715, 807, 913, 1033, 1166, 1313, 1483, 1667, 1875, 2111, 2369, 2655, 2977, 3332, 3729, 4170, 4657, 5195, 5797, 6459

Offset: 0

Gus Wiseman, Feb 15 2022

nonn

Partitions enumerated by this sequence include those in which all parts are either the same or distinct as well as partitions with an even number of parts all of which except one are the same. - Andrew Howroyd, Feb 15 2022

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (3111)    (4111)     (521)
                                     (111111)  (211111)   (2222)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A000005.
The version for normal multisets is 2^(n-1) - A283353(n-3).
The complement is counted by A351203, ranked by A351201.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A238130 and A238279 count compositions by number of runs.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A144300, A329746, A351291.

Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!UnsameQ@@Split[#]&]=={}&]],{n,0,15}]

\\ here Q(n) is A000009.
Q(n)={polcoef(prod(k=1, n, 1 + x^k + O(x*x^n)), n)}
a(n)={Q(n) + if(n, numdiv(n) - 1) + sum(k=1, (n-1)\3, sum(j=3, (n-1)\k, j%2==1 && n-k*j<>k))} \\ Andrew Howroyd, Feb 15 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 15 2022

A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694

Offset: 0

Gus Wiseman, Jan 24 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The a(5) = 1 through a(10) = 17 partitions:
  (311)  (411)   (511)    (422)     (522)      (622)
         (3111)  (4111)   (611)     (711)      (811)
                 (31111)  (3311)    (4221)     (4222)
                          (4211)    (4311)     (4411)
                          (5111)    (5211)     (5221)
                          (41111)   (6111)     (5311)
                          (311111)  (33111)    (6211)
                                    (42111)    (7111)
                                    (51111)    (42211)
                                    (411111)   (43111)
                                    (3111111)  (52111)
                                               (61111)
                                               (331111)
                                               (421111)
                                               (511111)
                                               (4111111)
                                               (31111111)

Allowing -1 gives A144300 = non-constant partitions.
Taking one of the two conditions gives A239955, ranked by A073492, A065201.
These partitions are ranked by A350841.
A000041 = integer partitions, strict A000009.
A034296 = flat (contiguous) partitions, strict A001227.
A073491 = numbers whose prime indices have no gaps, strict A137793.
A090858 = partitions with a single hole, ranked by A325284.
A116931 = partitions with differences != -1, strict A003114.
A116932 = partitions with differences != -1 or -2, strict A025157.
A277103 = partitions with the same number of odd parts as their conjugate.
A350837 = partitions with no adjacent doublings, strict A350840.
A350842 = partitions with differences != -2, strict A350844, sets A005314.
Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]

A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 11, 15, 20, 30, 38, 49, 65, 83, 108, 139, 178, 224, 286, 358, 437, 550, 684, 837, 1037, 1269, 1553, 1889, 2295, 2770, 3359, 4035, 4843, 5808, 6951, 8312, 9902, 11752, 13958, 16531, 19541, 23037, 27162, 31911, 37488, 43950, 51463, 60127, 70229

Offset: 0

Gus Wiseman, Mar 17 2023

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (22)   (32)    (33)    (43)
                   (31)   (41)    (42)    (52)
                   (211)  (221)   (51)    (61)
                          (311)   (321)   (322)
                          (2111)  (411)   (331)
                                  (2211)  (421)
                                  (3111)  (511)
                                          (2221)
                                          (3211)
                                          (4111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The complement is counted by A360254, ranks A360558.
These partitions have ranks A361395.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, reverse A058398.
A067538 counts partitions with integer mean, strict A102627.
A116608 counts partitions by number of distinct parts.
Cf. A106529, A144300, A237363, A237832, A240219, A316413, A349156.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),
     `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2023

Table[Length[Select[IntegerPartitions[n],2*Length[Union[#]]>=Length[#]&]],{n,0,30}]

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497

Offset: 0

Gus Wiseman, Feb 12 2022

nonn

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A144300.
The version for normal multisets is A283353.
The Heinz numbers of these partitions are A351201.
The complement is counted by A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions, ranked by A333489.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n,size=True):
        for q in permutations(Counter(p).elements(),s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

a(26) onwards from Andrew Howroyd, Jan 27 2024

A182114 Number of partitions of n with largest inscribed rectangle having area <= k; triangle T(n,k), 0<=n, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 5, 0, 0, 0, 1, 5, 7, 0, 0, 0, 0, 5, 7, 11, 0, 0, 0, 0, 3, 7, 13, 15, 0, 0, 0, 0, 1, 5, 16, 18, 22, 0, 0, 0, 0, 0, 3, 17, 21, 27, 30, 0, 0, 0, 0, 0, 1, 16, 22, 34, 38, 42, 0, 0, 0, 0, 0, 0, 13, 21, 39, 48, 54, 56

Offset: 0

Alois P. Heinz, Apr 12 2012

T(n,k) = A000041(k) for n

Sum_{n>=0} T(n,k) = A115725(k).

			T(5,4) = 5 because there are 5 partitions of 5 with largest inscribed rectangle having area <= 4: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4].
T(9,5) = 3: [1,1,1,2,4], [1,1,1,1,5], [1,1,2,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 3;
  0, 0, 0, 2, 5;
  0, 0, 0, 1, 5, 7;
  0, 0, 0, 0, 5, 7, 11;
  0, 0, 0, 0, 3, 7, 13, 15;
  0, 0, 0, 0, 1, 5, 16, 18, 22;
  0, 0, 0, 0, 0, 3, 17, 21, 27, 30;
  0, 0, 0, 0, 0, 1, 16, 22, 34, 38, 42;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Diagonal gives: A000041.
T(n,n-1) = A144300(n) = A000041(n) - A000005(n).
T(n+d,n) for d=2-10 give: A218623, A218624, A218625, A218626, A218627, A218628, A218629, A218630, A218631.
Cf. A115723, A115725.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i=1, `if`(t+n<=k, 1, 0), `if`(i<1, 0, b(n, i-1, t, k)+
       add(`if`(t+j<=k/i, b(n-i*j, i-1, t+j, k), 0), j=1..n/i))))
    end:
T:= (n, k)-> b(n, n, 0, k):
seq(seq(T(n, k), k=0..n), n=0..15);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i == 1, If[t + n <= k, 1, 0], If[i < 1, 0, b[n, i - 1, t, k] + Sum[If[t + j <= k/i, b[n - i*j, i - 1, t + j, k], 0], {j, 1, n/i}]]]] ; T[n_, k_] := b[n, n, 0, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=1..k} A115723(n,i) for n>0, T(0,0) = 1.

A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 50, 12, 14, 33, 54, 0, 115, 0, 92, 75, 31, 99, 323, 0, 45, 162, 443, 0, 507, 0, 467, 732, 88, 0, 1551, 274, 833, 627, 1228, 0, 2035, 1556, 2859, 1152, 221, 0, 9008, 0, 295, 4835, 5358

Offset: 1

Gus Wiseman, Jun 23 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(8) = 1 through a(18) = 12 partitions:
  3221  .  32221  .  4332    .  3222221  43332  5443      .  433332
                     5331       3322211  53331  6442         443331
                     322221     4222211  63321  7441         533322
                     422211                     32222221     533331
                                                33222211     543321
                                                42222211     633321
                                                52222111     733311
                                                             322222221
                                                             332222211
                                                             422222211
                                                             432222111
                                                             522222111

Non-constant partitions are counted by A144300, ranks A024619.
This is the non-constant case of A363719, ranks A363727.
These partitions have ranks A363729.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A363720, A363721.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A222970 Number of 1 X (n+1) 0..1 arrays with every row least squares fitting to a positive-slope straight line and every column least squares fitting to a zero- or positive-slope straight line, with a single point array taken as having zero slope.

Original entry on oeis.org

1, 2, 6, 12, 28, 54, 119, 230, 488, 948, 1979, 3860, 7978, 15624, 32072, 63014, 128746, 253588, 516346, 1019072, 2069590, 4091174, 8291746, 16412668, 33210428, 65808044, 132985161, 263755984, 532421062, 1056789662, 2131312530, 4233176854

Offset: 1

R. H. Hardin, Mar 10 2013

nonn

From Gus Wiseman, Jun 16 2023: (Start)
Also appears to be the number of integer compositions of n + 2 with weighted sum greater than reverse-weighted sum, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i, and the reverse is Sum_{i=1..k} i * y_{k-i+1}. The a(1) = 1 through a(4) = 12 compositions are:
  (21)  (31)   (32)    (42)
        (211)  (41)    (51)
               (221)   (231)
               (311)   (312)
               (1211)  (321)
               (2111)  (411)
                       (1311)
                       (2121)
                       (2211)
                       (3111)
                       (12111)
                       (21111)
The version for partitions is A144300, strict A111133.
(End)

			Some solutions for n=3:
  0 1 0 1    0 1 1 1    0 0 1 0    0 0 1 1    0 0 0 1

R. H. Hardin, Table of n, a(n) for n = 1..210

For >= instead of > we have A222855.
The case of equality is A222955.
Row 1 of A222969.
A053632 counts compositions by weighted sum (or  reverse-weighted sum).
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, reverse A318283.
Cf. A000005, A000041, A138364, A320387, A360672, A360675, A363626.

Showing 1-10 of 22 results. Next

cp's OEIS Frontend

A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

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A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

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A239955 Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).

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A351204 Number of integer partitions of n such that every permutation has all distinct runs.

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A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

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A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

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A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

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