A326622 -id:A326622 - cp's OEIS frontend

A067538 Number of partitions of n in which the number of parts divides n.

1, 2, 2, 4, 2, 8, 2, 11, 9, 14, 2, 46, 2, 24, 51, 66, 2, 126, 2, 202, 144, 69, 2, 632, 194, 116, 381, 756, 2, 1707, 2, 1417, 956, 316, 2043, 5295, 2, 511, 2293, 9151, 2, 10278, 2, 8409, 14671, 1280, 2, 36901, 8035, 21524, 11614, 25639, 2, 53138, 39810, 85004

Offset: 1

Naohiro Nomoto, Jan 27 2002

Also sum of p(n,d) over the divisors d of n, where p(n,m) is the count of partitions of n in exactly m parts. - Wouter Meeussen, Jun 07 2009
From Gus Wiseman, Sep 24 2019: (Start)
Also the number of integer partitions of n whose maximum part divides n. The Heinz numbers of these partitions are given by A326836. For example, the a(1) = 1 through a(8) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (321)                (431)
                                     (2211)               (2222)
                                     (3111)               (4211)
                                     (21111)              (22211)
                                     (111111)             (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
(End)

			a(3)=2 because 3 is a prime; a(4)=4 because the five partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}, and the number of parts in each of them divides 4 except for {2, 1, 1}.
From _Gus Wiseman_, Sep 24 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions whose length divides their sum are the following. The Heinz numbers of these partitions are given by A316413.
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (31)             (42)                 (53)
                    (1111)           (51)                 (62)
                                     (222)                (71)
                                     (321)                (2222)
                                     (411)                (3221)
                                     (111111)             (3311)
                                                          (4211)
                                                          (5111)
                                                          (11111111)
(End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Wouter Meeussen, n = 501..1000 from Alois P. Heinz, n = 1001..5000 from David A. Corneth)
Eric W. Weisstein, Partition Function P
Wikipedia, Integer Partition

Cf. A000005, A000041, A143773, A298422, A298423, A298426.
The strict case is A102627.
Partitions with integer geometric mean are A067539.
Cf. A018818, A102627, A316413, A326622, A326836, A326843, A326850.

Do[p = IntegerPartitions[n]; l = Length[p]; c = 0; k = 1; While[k < l + 1, If[ IntegerQ[ n/Length[ p[[k]] ]], c++ ]; k++ ]; Print[c], {n, 1, 57}, All]
p[n_,k_]:=p[n,k]=p[n-1,k-1]+p[n-k,k];p[n_,k_]:=0/;k>n;p[n_,n_]:=1;p[n_,0]:=0
Table[Plus @@ (p[n,# ]&/ @ Divisors[n]),{n,36}] (* Wouter Meeussen, Jun 07 2009 *)
Table[Count[IntegerPartitions[n], q_ /; IntegerQ[Mean[q]]], {n, 50}]  (*Clark Kimberling, Apr 23 2019 *)

a(n) = {my(nb = 0); forpart(p=n, if ((vecsum(Vec(p)) % #p) == 0, nb++);); nb;} \\ Michel Marcus, Jul 03 2018

# uses A008284_T
from sympy import divisors
def A067538(n): return sum(A008284_T(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Sep 21 2023

a(p) = 2 for all primes p.

Extended by Robert G. Wilson v, Oct 16 2002

A102627 Number of partitions of n into distinct parts in which the number of parts divides n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 4, 5, 1, 15, 1, 7, 14, 17, 1, 28, 1, 40, 28, 11, 1, 99, 31, 13, 49, 99, 1, 186, 1, 152, 76, 17, 208, 425, 1, 19, 109, 699, 1, 584, 1, 433, 823, 23, 1, 1625, 437, 1140, 193, 746, 1, 2003, 1748, 2749, 244, 29, 1, 7404, 1, 31, 4158, 3258, 3766, 6307, 1

Offset: 1

Vladeta Jovovic, Feb 01 2005

			From _Gus Wiseman_, Sep 24 2019: (Start)
The a(1) = 1 through a(12) = 15 strict integer partitions whose average is an integer (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)  (6)    (7)  (8)   (9)    (A)   (B)  (C)
                 (31)       (42)        (53)  (432)  (64)       (75)
                            (51)        (62)  (531)  (73)       (84)
                            (321)       (71)  (621)  (82)       (93)
                                                     (91)       (A2)
                                                                (B1)
                                                                (543)
                                                                (642)
                                                                (651)
                                                                (732)
                                                                (741)
                                                                (831)
                                                                (921)
                                                                (5421)
                                                                (6321)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

The BI-numbers of these partitions are given by A326669 (numbers whose binary indices have integer mean).
The non-strict case is A067538.
Strict partitions with integer geometric mean are A326625.
Strict partitions whose maximum divides their sum are A326850.
Cf. A018818, A033630, A316413, A326622, A326843, A326851.

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Sep 25 2019

npdp[n_]:=Count[Select[IntegerPartitions[n],Length[#]==Length[ Union[ #]]&], ?(Divisible[n,Length[#]]&)]; Array[npdp,70] (* _Harvey P. Dale, Feb 12 2016 *)
a[m_] := a[m] = Module[{b}, b[n_, i_, t_] := b[n, i, t] = If[i(i+1)/2 < n, 0, If[n == 0, If[Mod[m, t] == 0, 1, 0], b[n, i - 1, t] + b[n - i, Min[n - i, i - 1], t + 1]]]; If[PrimeQ[m], 1, b[m, m, 0]]];
Array[a, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

A359894 Number of integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 10, 13, 20, 28, 49, 53, 93, 113, 145, 203, 287, 329, 479, 556, 724, 955, 1242, 1432, 1889, 2370, 2863, 3502, 4549, 5237, 6825, 8108, 9839, 12188, 14374, 16958, 21617, 25852, 30582, 36100, 44561, 51462, 63238, 73386, 85990, 105272, 124729

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(4) = 1 through a(8) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)
         (311)   (3111)   (331)     (422)
         (2111)  (21111)  (421)     (431)
                          (511)     (521)
                          (2221)    (611)
                          (3211)    (4211)
                          (4111)    (5111)
                          (22111)   (22211)
                          (31111)   (32111)
                          (211111)  (41111)
                                    (221111)
                                    (311111)
                                    (2111111)

The complement is counted by A240219.
These partitions are ranked by A359890, complement A359889.
The odd-length case is ranked by A359892, complement A359891.
The odd-length case is A359896, complement A359895.
The strict case is A359898, complement A359897.
The odd-length strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284 and A058398 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
A359909 counts factorizations with the same mean as median, odd-len A359910.
Cf. A066571, A316313, A327472, A327482, A349156, A359906.

Table[Length[Select[IntegerPartitions[n],Mean[#]!=Median[#]&]],{n,0,30}]

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

a(n > 0) = A000041(n) - A067538(n).

A328966 Number of strict factorizations of n with integer average.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Nov 16 2019

nonn

			The a(n) factorizations for n = 2, 8, 24, 48, 96:
  (2)  (8)    (24)     (32)    (48)     (96)
       (2*4)  (4*6)    (4*8)   (6*8)    (2*48)
              (2*12)   (2*16)  (2*24)   (4*24)
              (2*3*4)          (4*12)   (6*16)
                               (2*4*6)  (8*12)
                                        (3*4*8)
                                        (2*3*16)
                                        (2*4*12)

The non-strict version is A326622.
Partitions with integer average are A067538.
Strict partitions with integer average are A102627.
Heinz numbers of partitions with integer average are A316413.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A051293, A078174, A078175, A326515, A326567/A326568, A326619/A326620, A326621, A326625.

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@@#&&IntegerQ[Mean[#]]&]],{n,2,100}]

A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite multisets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  1   1   0   0   0   0   0   0   0   0   0   0
   n=3:  1   1   1   0   0   0   0   0   0   0   0   0
   n=4:  1   1   1   2   0   0   0   0   0   0   0   0
   n=5:  1   1   1   2   1   1   0   0   0   0   0   0
   n=6:  1   1   1   2   1   2   0   2   1   0   0   0
   n=7:  1   1   1   2   1   2   1   2   1   1   0   2
   n=8:  1   1   1   2   1   2   1   3   1   1   0   3
   n=9:  1   1   1   2   1   2   1   3   2   1   0   3
  n=10:  1   1   1   2   1   2   1   3   2   2   0   3
  n=11:  1   1   1   2   1   2   1   3   2   2   1   3
  n=12:  1   1   1   2   1   2   1   3   2   2   1   4
For example, the A(9,12) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 1
   n+k=4: 0 0 1 1
   n+k=5: 0 0 0 1 1
   n+k=6: 0 0 0 1 1 1
   n+k=7: 0 0 0 0 1 1 1
   n+k=8: 0 0 0 0 2 1 1 1
   n+k=9: 0 0 0 0 0 2 1 1 1
  n+k=10: 0 0 0 0 0 1 2 1 1 1
  n+k=11: 0 0 0 0 0 1 1 2 1 1 1
  n+k=12: 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=13: 0 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=14: 0 0 0 0 0 0 2 1 2 1 2 1 1 1
  n+k=15: 0 0 0 0 0 0 1 2 1 2 1 2 1 1 1
  n+k=16: 0 0 0 0 0 0 0 1 3 1 2 1 2 1 1 1
For example, antidiagonal n+k=10 counts the following partitions:
  n=5: (5)
  n=6: (411), (2211)
  n=7: (31111)
  n=8: (2111111)
  n=9: (111111111)
so the 10th antidiagonal is: (0,0,0,0,0,1,2,1,1,1).

Row sums are A000041 = partitions of n, strict A000009, no ones A002865.
Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
Antidiagonal sums are A379667.
The case without ones is A379668, antidiagonal sums A379669 (zeros A379670).
The strict case is A379671, antidiagonal sums A379672.
The strict case without ones is A379678, antidiagonal sums A379679 (zeros A379680).
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A096765, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 19, 20, 21, 28, 29, 30, 31, 40, 41, 70, 71, 74, 75, 76, 77, 108, 123, 124, 211, 214, 215, 216, 217, 332, 333, 334, 335, 592, 593, 594, 595, 612, 613, 614, 615, 618, 639, 640, 641, 1160, 1183, 1324, 1325, 1328, 1329, 2176, 2177, 2196, 2197, 2198, 2199, 2414, 2415, 2416, 2443, 4000, 4001, 4002, 4003, 4006, 4007, 4008, 4009, 6626, 6627, 6628, 9753, 9756, 9757, 9758, 9759, 11136

Offset: 0

Gus Wiseman, Jul 14 2019

nonn

			The a(1) = 1 through a(9) = 19 subsets:
  {1}  {1}  {1}  {1}      {1}      {1}      {1}      {1}      {1}
       {2}  {2}  {2}      {2}      {2}      {2}      {2}      {2}
            {3}  {3}      {3}      {3}      {3}      {3}      {3}
                 {4}      {4}      {4}      {4}      {4}      {4}
                 {1,4}    {5}      {5}      {5}      {5}      {5}
                 {1,2,4}  {1,4}    {6}      {6}      {6}      {6}
                          {1,2,4}  {1,4}    {7}      {7}      {7}
                                   {1,2,4}  {1,4}    {8}      {8}
                                            {1,2,4}  {1,4}    {9}
                                                     {2,8}    {1,4}
                                                     {1,2,4}  {1,9}
                                                     {2,4,8}  {2,8}
                                                              {4,9}
                                                              {1,2,4}
                                                              {1,3,9}
                                                              {2,4,8}
                                                              {3,8,9}
                                                              {4,6,9}
                                                              {3,6,8,9}

Max Alekseyev, Table of n, a(n) for n = 0..255
Wikipedia, Geometric mean

First differences are A082553.
Partitions whose geometric mean is an integer are A067539.
Strict partitions whose geometric mean is an integer are A326625.
Subsets whose average is an integer are A051293.
Cf. A078174, A078175, A102627, A326567/A326568, A326622, A326623, A326624.

Table[Length[Select[Subsets[Range[n]],IntegerQ[GeometricMean[#]]&]],{n,0,10}]

a(n) = A357413(n) + A357414(n). For a squarefree n, a(n) = a(n-1) + 1. - Max Alekseyev, Mar 01 2025

Terms a(57) onward from Max Alekseyev, Mar 01 2025

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A379671 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers with sum n and product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite sets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   1   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   1   1   0   0   0   0   0   0   0   0
   n=5:  0   0   0   1   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   1   2   0   1   0   0   0   0
   n=7:  0   0   0   0   0   1   1   1   0   1   0   1
   n=8:  0   0   0   0   0   0   1   1   0   1   0   2
   n=9:  0   0   0   0   0   0   0   1   1   0   0   1
  n=10:  0   0   0   0   0   0   0   0   1   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   1   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   1   1
The A(8,12) = 2 sets are: {2,6}, {1,3,4}.
The A(14,40) = 2 sets are: {4,10}, {1,5,8}.
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 1 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 1 0 0
   n+k=8: 0 0 0 0 1 0 0 0
   n+k=9: 0 0 0 0 0 1 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 1 0 0 0 0
  n+k=12: 0 0 0 0 0 0 2 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=11 counts the following sets:
  n=5: {2,3}
  n=6: {1,5}
so the 11th antidiagonal is: (0,0,0,0,0,1,1,0,0,0,0).

Row sums are A000009 = strict partitions, non-strict A000041.
Column sums are 2*A045778 where A045778 = strict factorizations, non-strict A001055.
Antidiagonal sums are A379672, non-strict A379667 (zeros A379670).
Without ones we have A379678, antidiagonal sums A379679 (zeros A379680).
The non-strict version is A379666, without ones A379668.
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A002865, A003963, A028422, A069016, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 15 2019

nonn

First differs from A294336 and A316782 at a(36) = 5.

			The a(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).
  (4)    (8)      (9)    (16)       (25)   (27)     (32)         (36)
  (2*2)  (2*2*2)  (3*3)  (2*8)      (5*5)  (3*3*3)  (2*2*2*2*2)  (4*9)
                         (4*4)                                   (6*6)
                         (2*2*2*2)                               (2*18)
                                                                 (3*12)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Wikipedia, Geometric mean
Index entries for sequences computed from exponents in factorization of n

Positions of terms > 1 are the perfect powers A001597.
Partitions with integer geometric mean are A067539.
Subsets with integer geometric mean are A326027.
Factorizations with integer average and geometric mean are A326647.
Cf. A001055, A082553, A322794, A326514, A326515, A326516, A326622, A326623, A326624, A326625.

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],IntegerQ[GeometricMean[#]]&]],{n,2,100}]

A326028(n, m=n, facmul=1, facnum=0) = if(1==n,facnum>0 && ispower(facmul,facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326028(n/d, d, facmul*d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

a(2^n) = A067538(n).

a(89) onwards from Antti Karttunen, Nov 10 2024

cp's OEIS Frontend

A067538 Number of partitions of n in which the number of parts divides n.

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A102627 Number of partitions of n into distinct parts in which the number of parts divides n.

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A359894 Number of integer partitions of n whose parts do not have the same mean as median.

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A349156 Number of integer partitions of n whose mean is not an integer.

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A328966 Number of strict factorizations of n with integer average.

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A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

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A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

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A348551 Heinz numbers of integer partitions whose mean is not an integer.

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