A002865 -id:A002865 - cp's OEIS frontend

A362611 Number of modes in the prime factorization of n.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2

Offset: 1

Gus Wiseman, May 05 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}.

a(n) depends only on the prime signature of n. - Andrew Howroyd, May 08 2023

			The factorization of 450 is 2*3*3*5*5, modes {3,5}, so a(450) = 2.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so a(900) = 3.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so a(1500) = 1.
The factorization of 8820 is 2*2*3*3*5*7*7, modes {2,3,7}, so a(8820) = 3.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Positions of first appearances are A002110.
Positions of 1's are A356862, counted by A362608.
Positions of terms > 1 are A362605, counted by A362607.
For co-mode we have A362613, counted by A362615.
This statistic (mode-count) has triangular form A362614.
A027746 lists prime factors (with multiplicity).
A112798 lists prime indices, length A001222, sum A056239.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362606 ranks partitions with more than one co-mode, counted by A362609.
Cf. A000040, A000720, A001221, A002865, A072774, A215366, A327473, A327476, A359908.

Table[x=Last/@If[n==1,0,FactorInteger[n]];Count[x,Max@@x],{n,100}]

a(n) = if(n==1, 0, my(f=factor(n)[,2], m=vecmax(f)); #select(v->v==m, f)) \\ Andrew Howroyd, May 08 2023

from sympy import factorint
def A362611(n): return list(v:=factorint(n).values()).count(max(v,default=0)) # Chai Wah Wu, May 08 2023

For n > 1, 1 <= a(n) << log n. - Charles R Greathouse IV, May 09 2023

a(n) <= A001221(n), with equality if and only if n is a power of a squarefree number (A072774). - Amiram Eldar, Mar 02 2025

A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 5, 3, 4, 4, 4, 4, 4, 4, 4, 7, 5, 7, 8, 7, 5, 7, 8, 7, 7, 8, 8, 7, 7, 8, 12, 9, 12, 9, 17, 9, 12, 9, 12, 14, 11, 12, 12, 13, 13, 12, 12, 11, 14, 21, 15, 19, 15, 21, 24, 21, 15, 19, 15, 21, 24, 19, 20, 19, 21, 22, 22, 21, 19, 20, 19, 24, 34, 23, 30, 24, 30, 25, 46, 25, 30, 24, 30, 23, 34

Offset: 2

N. J. A. Sloane

			For n = 4 there are two partitions (4, 2+2) with no subsum equal to 1, two (4, 3+1) with no subsum equal to 2 and two (4, 2+2) with no subsum equal to 3.
Triangle T(n,k) begins:
   1;
   1,  1;
   2,  2,  2;
   2,  2,  2,  2;
   4,  3,  5,  3,  4;
   4,  4,  4,  4,  4,  4;
   7,  5,  7,  8,  7,  5,  7;
   8,  7,  7,  8,  8,  7,  7,  8;
  12,  9, 12,  9, 17,  9, 12,  9, 12;
  ...
From _Gus Wiseman_, Oct 11 2023: (Start)
Row n = 8 counts the following partitions:
  (8)     (8)    (8)     (8)     (8)     (8)    (8)
  (62)    (71)   (71)    (71)    (71)    (71)   (62)
  (53)    (53)   (62)    (62)    (62)    (53)   (53)
  (44)    (44)   (611)   (611)   (611)   (44)   (44)
  (422)   (431)  (44)    (53)    (44)    (431)  (422)
  (332)          (422)   (521)   (422)          (332)
  (2222)         (2222)  (5111)  (2222)         (2222)
                         (332)
(End)

Alois P. Heinz, Rows n = 2..98, flattened
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Column k = 0 and diagonal k = n are both A002865.
Central diagonal n = 2k is A006827.
The complement with expanded domain is A365543.
The strict case is A365663, complement A365661.
Row sums are A365918, complement A304792.
For subsets instead of partitions we have A366320, complement A365381.
A000041 counts integer partitions, strict A000009.
A276024 counts distinct subset-sums of partitions.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000124, A002219, A093971, A108917, A122768, A299701, A365658.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(0 in s or n in s, 0, `if`(n=0 or s={}, g(n, i),
     `if`(i<1, 0, b(n, i-1, s)+`if`(i>n, 0, b(n-i, i,
      select(y-> 0<=y and y<=n-i, map(x-> [x, x-i][], s)))))))
    end:
T:= (n, k)-> b(n, n, {min(k, n-k)}):
seq(seq(T(n, k), k=1..n-1), n=2..16);  # Alois P. Heinz, Jul 13 2012

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i > 1, g[n, i-1], 0] + If[i > n, 0, g[n-i, i]]]; b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0 || s == {}, g[n, i], If[i < 1, 0, b[n, i-1, s] + If[i > n, 0, b[n-i, i, Select[Flatten[s /. x_ :> {x, x-i}], 0 <= # <= n-i &]]]]]]; t[n_, k_] := b[n, n, {Min[k, n-k]}]; Table[t[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Aug 20 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#],k]&]],{n,2,10},{k,1,n-1}] (* Gus Wiseman, Oct 11 2023 *)

Corrected and extended by Don Reble, Nov 04 2001

A319005 Number of integer partitions of n whose product of parts is >= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 7, 13, 18, 28, 40, 60, 80, 113, 152, 205, 266, 353, 454, 590, 751, 959, 1210, 1529, 1905, 2381, 2953, 3658, 4501, 5539, 6772, 8278, 10065, 12230, 14801, 17893, 21544, 25921, 31089, 37240, 44478, 53068, 63150, 75063, 89018, 105438, 124632

Offset: 0

Gus Wiseman, Oct 22 2018

nonn

			The a(1) = 1 through a(9) = 18 partitions:
  (1)  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                 (22)  (32)  (33)   (43)    (44)     (54)
                             (42)   (52)    (53)     (63)
                             (222)  (322)   (62)     (72)
                             (321)  (331)   (332)    (333)
                                    (421)   (422)    (432)
                                    (2221)  (431)    (441)
                                            (521)    (522)
                                            (2222)   (531)
                                            (3221)   (621)
                                            (3311)   (3222)
                                            (4211)   (3321)
                                            (22211)  (4221)
                                                     (4311)
                                                     (5211)
                                                     (22221)
                                                     (32211)
                                                     (33111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Pankaj Jyoti Mahanta, On the number of partitions of n whose product of the summands is at most n, arXiv:2010.07353 [math.CO], 2020.

Column sums of A319000.

Cf. A001055, A002865, A069016, A096276, A301987, A318950, A319057, A319916.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, `if`(p>1,
      0, 1), b(n, i-1, p) +b(n-i, min(i, n-i), max(p/i, 1)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 22 2018

Table[Length[Select[IntegerPartitions[n],Times@@#>=n&]],{n,50}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[p > 1, 0, 1],
     b[n, i - 1, p] + b[n - i, Min[i, n - i], Max[p/i, 1]]];
a[n_] := b[n, n, n];
a /@ Range[0, 50] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 11, 3, 1, 0, 16, 4, 2, 0, 21, 6, 3, 0, 29, 8, 4, 1, 0, 43, 7, 5, 1, 0, 54, 13, 8, 2, 0, 78, 12, 8, 3, 0, 102, 17, 11, 5, 0, 131, 26, 12, 6, 1, 0, 175, 29, 17, 9, 1, 0, 233, 33, 18, 11, 2, 0, 295, 47, 25

Offset: 0

Gus Wiseman, May 04 2023

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}.

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  11   3   1
   0  16   4   2
   0  21   6   3
   0  29   8   4   1
   0  43   7   5   1
   0  54  13   8   2
   0  78  12   8   3
   0 102  17  11   5
   0 131  26  12   6   1
   0 175  29  17   9   1
Row n = 8 counts the following partitions:
  (8)         (53)    (431)
  (44)        (62)    (521)
  (332)       (71)
  (422)       (3311)
  (611)
  (2222)
  (3221)
  (4211)
  (5111)
  (22211)
  (32111)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Alois P. Heinz, Rows n = 0..800, flattened

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
Column k = 1 is A362608, ranks A356862.
This statistic (mode-count) is ranked by A362611.
For co-modes we have A362615, ranked by A362613.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.

msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024

A054973 Number of numbers whose divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 1, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 3, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 1, 0, 1, 0, 0, 4, 0

Offset: 1

Henry Bottomley, May 16 2000

nonn

a(n) = frequency of values n in A000203(m), where A000203(m) = sum of divisors of m. a(n) >= 1 for such n that A175192(n) = 1, a(n) >= 1 if A000203(m) = n for any m. a(n) = 0 for such n that A175192(n) = 0, a(n) = 0 if A000203(m) = n has no solution. - Jaroslav Krizek, Mar 01 2010
First occurrence of k: 2, 1, 12, 24, 96, 72, ..., = A007368. - Robert G. Wilson v, May 14 2014
a(n) is also the number of positive terms in the n-th row of triangle A299762. - Omar E. Pol, Mar 14 2018
Also the number of integer partitions of n whose parts form the set of divisors of some number (necessarily the greatest part). The Heinz numbers of these partitions are given by A371283. For example, the a(24) = 3 partitions are: (23,1), (15,5,3,1), (14,7,2,1). - Gus Wiseman, Mar 22 2024

			a(12) = 2 since 11 has factors 1 and 11 with 1 + 11 = 12 and 6 has factors 1, 2, 3 and 6 with 1 + 2 + 3 + 6 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203 (sum-of-divisors function).
For partial sums see A074753.
Cf. A002191, A007609, A308039.
The non-strict version is A371284, ranks A371288.
These partitions have ranks A371283, unsorted version A275700.
A000005 counts divisors, row-lengths of A027750.
A000041 counts integer partitions, strict A000009.
Cf. A001221, A002865, A008289, A371286, A371285.

nn = 105; t = Table[0, {nn}]; k = 1; While[k < 6 nn^(3/2)/Pi^2, d = DivisorSigma[1, k]; If[d < nn + 1, t[[d]]++]; k++]; t (* Robert G. Wilson v, May 14 2014 *)
Table[Length[Select[IntegerPartitions[n],#==Reverse[Divisors[Max@@#]]&]],{n,30}] (* Gus Wiseman, Mar 22 2024 *)

a(n)=v = vector(0); for (i = 1, n, if (sigma(i) == n, v = concat(v, i));); #v; \\ Michel Marcus, Oct 22 2013

a(n)=sum(k=1,n,sigma(k)==n) \\ Charles R Greathouse IV, Nov 12 2013

first(n)=my(v=vector(n),t); for(k=1,n, t=sigma(n); if(t<=n, v[t]++)); v \\ Charles R Greathouse IV, Mar 08 2017

A054973(n)=#invsigma(n) \\ See Alekseyev link for invsigma(). - M. F. Hasler, Nov 21 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A308039. - Amiram Eldar, Dec 23 2024

Incorrect comment deleted by M. F. Hasler, Nov 21 2019

A237668 Number of partitions of n such that some part is a sum of two or more other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023

			a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (3111)   (31111)   (3221)     (4311)
                             (21111)  (211111)  (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (33111)
                                                (41111)    (42111)
                                                (221111)   (222111)
                                                (311111)   (321111)
                                                (2111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 0..100
Giovanni Resta, C program for computing a(0)-a(100)

Cf. A179009.
The binary complement is A236912, ranks A364461.
The binary version is A237113, ranks A364462.
The complement is counted by A237667, ranks A364531.
The binary version with re-usable parts is A363225, ranks A364348.
The strict case is A364272.
The binary complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364532.
For subsets instead of partitions we have A364534, complement A151897.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts distinct subset-sums of prime indices.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.

z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
(* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,15}] (* Gus Wiseman, Aug 12 2023 *)

a(21)-a(47) from Giovanni Resta, Feb 22 2014

A138879 Sum of all parts of the last section of the set of partitions of n.

Original entry on oeis.org

1, 3, 5, 11, 15, 31, 39, 71, 94, 150, 196, 308, 389, 577, 750, 1056, 1353, 1881, 2380, 3230, 4092, 5412, 6821, 8935, 11150, 14386, 17934, 22834, 28281, 35735, 43982, 55066, 67551, 83821, 102365, 126267, 153397, 188001, 227645, 277305, 334383

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Row sums of the triangles A135010, A138121, A138151 and others related to the section model of partitions (see A135010 and A138121).
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and A002865.
Convolution of A340793 and A000041.
Row sums of triangles A339278, A340426, A340583. (End)
a(n) is also the sum of all divisors of all terms of n-th row of A336811. These divisors are also all parts in the last section of the set of partitions of n. - Omar E. Pol, Jul 27 2021
Row sums of A336812. - Omar E. Pol, Aug 03 2021

			a(6)=31 because the parts of the last section of the set of partitions of 6 are (6), (3,3), (4,2), (2,2,2), (1), (1), (1), (1), (1), (1), (1), so the sum is a(6) = 6 + 3 + 3 + 4 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 31.
From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    3      5        11         15           31
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the sum of all divisors of these terms is [1 + 2 + 3 + 6] + [1 + 2 + 4] + [1 + 3] + [1 + 2] + [1 + 2] + [1] + [1] = 31, so a(6) = 31. - _Omar E. Pol_, Jul 27 2021

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A000203, A002865, A066186, A133041, A135010, A138121, A138135 - A138138, A138151, A138880, A139100, A237593, A336811, A336812, A338156, A339278, A340035, A340426, A340583, A340793.

A066186 := proc(n) n*combinat[numbpart](n) ; end proc:
A138879 := proc(n) A066186(n)-A066186(n-1) ; end proc:
seq(A138879(n),n=1..80) ; # R. J. Mathar, Jan 27 2011

Table[PartitionsP[n]*n - PartitionsP[n-1]*(n-1), {n, 1, 50}] (* Vaclav Kotesovec, Oct 21 2016 *)

for(n=1, 50, print1(numbpart(n)*n - numbpart(n - 1)*(n - 1),", ")) \\ Indranil Ghosh, Mar 19 2017

from sympy.ntheory import npartitions
print([npartitions(n)*n - npartitions(n - 1)*(n - 1) for n in range(1, 51)]) # Indranil Ghosh, Mar 19 2017

a(n) = A000041(n)*n - A000041(n-1)*(n-1) = A138880(n) + A000041(n-1).
a(n) = A066186(n) - A066186(n-1), for n>=1.
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi/(12*sqrt(2*n)) * (1 - (72 + 13*Pi^2) / (24*Pi*sqrt(6*n)) + (7/12 + 3/(2*Pi^2) + 217*Pi^2/6912)/n - (15*sqrt(3/2)/(16*Pi) + 115*Pi/(288*sqrt(6)) + 4069*Pi^3/(497664*sqrt(6)))/n^(3/2)). - Vaclav Kotesovec, Oct 21 2016, extended Jul 06 2019
G.f.: x*(1 - x)*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

a(34) corrected by R. J. Mathar, Jan 27 2011

A336811 Irregular triangle read by rows T(n,k) in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive integers A000027, with n >= 1 and k >= 1.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 3, 2, 1, 1, 6, 4, 3, 2, 2, 1, 1, 7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 10, 8, 7, 6, 6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 20 2020

In other words: row n lists A028310(n-1) blocks where the m-th block consists of A187219(m) copies of n - m + [m=1], with n >= 1 and m >= 1, where [] is the Iverson bracket. [Corrected by Paolo Xausa, Feb 10 2023]
All divisors of all terms in row n are also all parts in the last section of the set of partitions of n.
Thus all divisors of all terms of the first n rows of triangle are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n. - Omar E. Pol, Jun 19 2021
From Omar E. Pol, Jul 31 2021: (Start)
The number of k's in row n is equal to A002865(n-k), 1 <= k <= n.
The number of terms >= k in row n is equal to A000041(n-k), 1 <= k <= n.
The number of k's in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000041(n-k), 1 <= k <= n.
The number of terms >= k in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.
First n rows of triangle (or first A000070(n-1) terms of the sequence) in nonincreasing order give the n-th row of A176206. (End)

			Triangle begins:
1;
2;
3, 1;
4, 2, 1;
5, 3, 2, 1, 1;
6, 4, 3, 2, 2, 1, 1;
7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1;
8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1;
9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
...
For n = 6, by definition the length of row 6 is A000041(6-1) = A000041(5) = 7, so the row 6 of triangle has seven terms. Since every column lists the positive integers A000027 so the row 6 is [6, 4, 3, 2, 2, 1, 1].
Then we have that the divisors of the numbers of the 6th row are:
.
6th row of the triangle ---------->   6 4 3 2 2 1 1
                                      3 2 1 1 1
                                      2 1
                                      1
.
There are seven 1's, four 2's, two 3's, one 4 and one 6.
In total there are 7 + 4 + 2 + 1 + 1 = 15 divisors.
On the other hand the last section of the set of the partitions of 6 can be represented in several ways, five of them as shown below:
._ _ _ _ _ _
|_ _ _      |       6    6                  6                       6
|_ _ _|_    |     3 3    3 3              3   3                     3   3
|_ _    |   |     4 2    4 2            4       2                     4     2
|_ _|_ _|_  |   2 2 2    2 2 2        2   2       2                 2 2   2
          | |       1      1                        1                           1
          | |       1        1                        1                       1
          | |       1        1                          1                   1
          | |       1          1                          1               1
          | |       1          1                            1           1
          | |       1            1                            1       1
          |_|       1              1                            1   1
.
   Figure 1.  Figure 2.  Figure 3.        Figure 4.                   Figure 5.
.
In every figure there are seven 1's, four 2's, two 3's, one 4 and one 6, as shown also the 6th row of A182703.
In total there are 7 + 4 + 2 + 1 + 1 = A138137(6) = 15 parts in every figure.
Figure 5 is an arrangement that shows the correspondence between divisors and parts since the columns give the divisors of the terms of 6th row of triangle.
Finally we can see that all divisors of all numbers in the 6th row of the triangle are the same positive integers as all parts in the last section of the set of the partitions of 6.
Example edited by _Omar E. Pol_, Aug 10 2021

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened).

Row sums give A000070.
Row n has length A000041(n-1).
Every column k gives A000027.
Companion of A176206.
Cf. A000007, A000041, A027750, A028310, A002865, A133735, A135010, A138121, A138137, A182703, A187219, A207378, A221529, A336812, A339278, A340035, A340061, A346741.

A336811[row_]:=Flatten[Table[ConstantArray[row-m,PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];
Array[A336811,10] (* Generates 10 rows *) (* Paolo Xausa, Feb 10 2023 *)

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--;); 1+s;}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1, 22, 4, 2, 1, 0, 0, 0, 0, 1, 30, 7, 2, 1, 1, 0, 0, 0, 0, 1, 42, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Clark Kimberling

At least one part is k and each part is at least k.
From Emeric Deutsch, Feb 19 2006: (Start)
Also number of partitions of n in which the largest part occurs exactly k times. Example: T(6,2)=2 because we have [3,3] and [2,2,1,1].
G.f. of column k is x^k/prod(j>=k, 1-x^j ) (k>=1).
Row sums yield the partition numbers (A000041).
T(n,1) = A000041(n-1) (the partition numbers).
T(n,2) = A002865(n-2) (n>=2).
T(n,3)=A026796(n). T(n,4) = A026797(n). T(n,5) = A026798(n). T(n,6) = A026799(n). T(n,7) = A026800(n). T(n,8) = A026801(n). T(n,9) = A026802(n). T(n,10) = A026803(n).
Sum(k*T(n,k),k=1..n) = A046746(n). (End)
Triangle inverse = A161363. - Gary W. Adamson, Jun 07 2009
T(n,g) is also the number of not necessarily connected 2-regular graphs with girth exactly g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012
From Bob Selcoe, Jul 24 2014 (Start):
Below is a process to generate equations for column k.
Let P be the partition numbers A000041(n-j) and let f(k) denote equations which generate column k.
To find f(k), start with f(1) = P(n-j), j=1. Thus T(n,1) = f(1) = P(n-1). This is the equation for column 1.
To find f(k) k>1, first sum the terms of f(k-1) replacing the value j with j+1, and then subtract the terms of f(k-1) replacing the value j with j+k. So to find f(2) (i.e., the equation for column 2, where k=2), start with f(1) = P(n-1); first replace j with j+1 (yielding P(n-2)), and then replace j with j+2 (yielding P(n-3)).  Subtracting the second term from the first, we get: f(2) = P(n-2) - P(n-3).
To find f(3), start with f(2), replace j with j+1 (yielding P(n-3) - P(n-4)) and then replace j with j+3 (yielding P(n-5) - P(n-6)). Subtracting the second group of terms from the first, we get: f(3) = P(n-3) - P(n-4) - P(n-5) + P(n-6). This is the equation for column 3; also the equation for T(n,3) = A026796(n). So for example, T(13,3) = 5 because P(13-3) - P(13-4) - P(13-5) + P(13-6) = 42 - 30 - 22 + 15 = 5.
Continue as above to find f(k) k={4..inf.}.  This will generate equations for T(n,4) = A026797(n), T(n,5) = A026798(n), T(n,6) = A026799(n), ad inf.
(End)

			T(12,3) = 4 because we have [9,3], [6,3,3], [5,4,3] and [3,3,3,3]. - Edited by _Bob Selcoe_, Sep 03 2016
Triangle starts:
    1;
    1,  1;
    2,  0, 1;
    3,  1, 0, 1;
    5,  1, 0, 0, 1;
    7,  2, 1, 0, 0, 1;
   11,  2, 1, 0, 0, 0, 1;
   15,  4, 1, 1, 0, 0, 0, 1;
   22,  4, 2, 1, 0, 0, 0, 0, 1;
   30,  7, 2, 1, 1, 0, 0, 0, 0, 1;
   42,  8, 3, 1, 1, 0, 0, 0, 0, 0, 1;
   56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1;
   77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  135, 24, 9, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Kevin Brown, On Euler's Pentagonal Theorem, 1994-2008.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g.
Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No. 1, December 2008. pp. 176-187.
Tilman Piesk, First 50 rows and illustrations of columns n = 2, 3, 4, 5, 6, 7, 8. a(n, k) is the number of gray fields in row n of table k.

Row sums give A000041.
Cf. A000041, A046746, A008284, A161363, A161364, A238341.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: this sequence (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012

g:=sum(t^i*x^i/product(1-x^j,j=i..30),i=1..30): gser:=simplify(series(g,x=0,19)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # Emeric Deutsch, Feb 19 2006
nmax:=13; for n from 1 to nmax do T(n,n):=1 od: for n from 1 to nmax do for k from floor(n/2)+1 to n-1 do T(n,k):=0 od: od: for n from 2 to nmax do for k from 1 to floor(n/2) do T(n,k):=sum(T(n-k,i),i=k..n-k) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
nmax:=13; with(combinat): for n from 1 to nmax do for k from n+1 to nmax do T(n,k):=0 od: od: for n from 1 to nmax do T(n,1):=numbpart(n-1) od: for n from 1 to nmax do T(n,n):=1 od: for n from 2 to nmax do for k from 2 to n-1 do T(n,k) := T(n-1,k-1) - T(n-k,k-1) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
#
p:= (f, g)-> zip((x,y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local h;
      h:= `if`(n=i and i>0, [0$(i-1), 1], []);
      `if`(i<1, h, p(p(h, b(n, i-1)), `if`(n b(n, n)[]:
seq(T(n), n=1..14); # Alois P. Heinz, Mar 28 2012

t[n_, k_] /; k<1 || k>n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[t[n-k, i], {i, k, n-k}]; Flatten[ Table[t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, May 11 2012, after PARI *)

{T(n, k) = if( k<1 || k>n, 0, if( n==k, 1, sum(i=k, n-k, T(n-k, i))))} \\ Michael Somos, Feb 06 2003

A026794(n,k)=#select(p->p[1]==k,partitions(n,[k,n])) \\ For illustration: Creates the list of all partitions of n with smallest part equal to k. - M. F. Hasler, Jun 14 2018

T(n, k) = sum{T(n-k, i), k<=i<=n-k} for k=1, 2, ..., m, T(n, k)=0 for k=m+1, ..., n-1, where m=floor(n/2); T(n, n)=1 for n >= 1.
G.f.: G(t,x)=sum(t^i*x^i/product(1-x^j, j=i..infinity), i=1..infinity). - Emeric Deutsch, Feb 19 2006
G.f.: Sum_{k>=1} tx^k/(1-tx^k)/product(1-x^j,j=1..k-1). - Emeric Deutsch, Mar 13 2006
T(n,k) = T(n-1,k-1) - T(n-k,k-1) for n>=2 and 2<=k<=(n-1) with T(n,1) = A000041(n-1), T(n,n) = 1 for n>=1 and T(n,k) = 0 for k>n. - Johannes W. Meijer, Jun 21 2010
T(k,k) = 1 and T(n,1) = row sum (n-1); thus Meijer's 2010 formula generates the triangle without a priori reference to A000041 (the partition sequence). - Bob Selcoe, Sep 03 2016

More terms from Emeric Deutsch, Feb 19 2006

A363225 Number of integer partitions of n containing three parts (a,b,c) (repeats allowed) such that a + b = c. A variation of sum-full partitions.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 29, 43, 58, 81, 109, 148, 195, 263, 339, 445, 574, 744, 942, 1209, 1515, 1923, 2399, 3005, 3721, 4629, 5693, 7024, 8589, 10530, 12804, 15596, 18876, 22870, 27538, 33204, 39816, 47766, 57061, 68161, 81099, 96510, 114434, 135634

Offset: 0

Gus Wiseman, Jul 19 2023

nonn

Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (432)
                       (2211)   (3211)    (521)      (621)
                       (21111)  (22111)   (3221)     (3321)
                                (211111)  (4211)     (4221)
                                          (22211)    (4311)
                                          (32111)    (5211)
                                          (221111)   (22221)
                                          (2111111)  (32211)
                                                     (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

For subsets of {1..n} we have A093971, A088809 without re-using parts.
The complement for subsets is A007865, A085489 without re-using parts.
Without re-using parts we have A237113, complement A236912.
For sums of any length > 1 (without re-usable parts) we have A237668, complement A237667.
The strict case is A363226.
The complement is counted by A364345, strict A364346.
These partitions have ranks A364348, complement A364347.
The strict linear combination-free version is A364350.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A108917, A237984, A326083, A363260.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,15}]

from collections import Counter
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A363225(n): return sum(1 for p in partitions(n) if any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 21 2023

a(31)-a(48) from Chai Wah Wu, Sep 21 2023

cp's OEIS Frontend

A362611 Number of modes in the prime factorization of n.

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A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

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A319005 Number of integer partitions of n whose product of parts is >= n.

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A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

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A054973 Number of numbers whose divisors sum to n.

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A237668 Number of partitions of n such that some part is a sum of two or more other parts.

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A138879 Sum of all parts of the last section of the set of partitions of n.

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