A067539 -id:A067539 - 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

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).

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[#]]]&]

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

A326623 Heinz numbers of integer partitions whose geometric mean is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 42, 43, 46, 47, 49, 53, 57, 59, 61, 64, 67, 71, 73, 76, 79, 81, 83, 89, 97, 101, 103, 106, 107, 109, 113, 121, 125, 126, 127, 128, 131, 137, 139, 149, 151, 157, 161, 163, 167, 169

Offset: 1

Gus Wiseman, Jul 14 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Wikipedia, Geometric mean

The enumeration of these partitions by sum is given by A067539.
Heinz numbers of partitions with integer average are A316413.
The case without prime powers is A326624.
Subsets whose geometric mean is an integer are A326027.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A078175, A102627, A326567/A326568, A326622, A326625.

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

A326625 Number of strict integer partitions of n whose geometric mean is an integer.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 2, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 5, 2, 3, 3, 3, 5, 1, 3, 5, 5, 3, 4, 4, 7, 7, 5, 5, 2, 4, 2, 5, 7, 4, 6, 9, 5, 7, 7, 8, 7, 5, 11, 5, 9, 9, 9, 7, 9, 5, 13, 7, 9, 7, 11, 12, 7, 7, 12, 9, 13, 11, 10, 13, 7, 14

Offset: 0

Gus Wiseman, Jul 14 2019

nonn

			The a(63) = 9 partitions:
  (63)  (36,18,9)  (54,4,3,2)   (36,18,6,2,1)   (36,9,8,6,3,1)
        (48,12,3)  (27,24,8,4)  (18,16,12,9,8)
                   (32,18,9,4)
The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5)
   5: (4,1)
   6: (6)
   7: (7)
   7: (4,2,1)
   8: (8)
   9: (9)
  10: (10)
  10: (9,1)
  10: (8,2)
  11: (11)
  12: (12)
  13: (13)
  13: (9,4)
  13: (9,3,1)
  14: (14)
  14: (8,4,2)
  15: (15)
  15: (12,3)
  16: (16)

Wikipedia, Geometric mean

Partitions whose geometric mean is an integer are A067539.
Strict partitions whose average is an integer are A102627.
Cf. A078174, A078175, A326027, A326567/A326568, A326623, A326624.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,30}]

A326641 Number of integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 2, 7, 2, 4, 5, 6, 2, 6, 2, 10, 6, 4, 2, 11, 4, 6, 5, 8, 2, 15, 2, 10, 6, 6, 8, 16, 2, 4, 8, 20, 2, 17, 2, 8, 17, 4, 2, 27, 9, 20, 8, 14, 2, 21, 10, 35, 10, 6, 2, 48, 2, 4, 41, 39, 12, 28, 2, 17, 10, 64, 2, 103, 2, 6, 23

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

The Heinz numbers of these partitions are given by A326645.

			The a(4) = 3 through a(10) = 6 partitions (A = 10):
  (4)     (5)      (6)       (7)        (8)         (9)          (A)
  (22)    (11111)  (33)      (1111111)  (44)        (333)        (55)
  (1111)           (222)                (2222)      (111111111)  (82)
                   (111111)             (11111111)               (91)
                                                                 (22222)
                                                                 (1111111111)

Wikipedia, Geometric mean

Partitions with integer mean are A067538.
Partitions with integer geometric mean are A067539.
Non-constant partitions with integer mean and geometric mean are A326642.
Subsets with integer mean and geometric mean are A326643.
Heinz numbers of partitions with integer mean and geometric mean are A326645.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A051293, A078175, A082553, A102627, A316413, A326027, A326623, A326644, A326646, A326647.

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

A360241 Number of integer partitions of n whose distinct parts have integer mean.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 6, 13, 13, 22, 19, 43, 34, 56, 66, 97, 92, 156, 143, 233, 256, 322, 341, 555, 542, 710, 831, 1098, 1131, 1644, 1660, 2275, 2484, 3035, 3492, 4731, 4848, 6063, 6893, 8943, 9378, 12222, 13025, 16520, 18748, 22048, 24405, 31446, 33698, 41558

Offset: 0

Gus Wiseman, Feb 02 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (511)      (53)
                    (1111)           (51)      (3211)     (62)
                                     (222)     (31111)    (71)
                                     (321)     (1111111)  (422)
                                     (3111)               (2222)
                                     (111111)             (3221)
                                                          (3311)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition (32111) has distinct parts {1,2,3} with mean 2, so is counted under a(8).

For parts instead of distinct parts we have A067538, ranked by A316413.
The strict case is A102627.
These partitions are ranked by A326621.
For multiplicities instead of distinct parts: A360069, ranked by A067340.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, also A327482.
A116608 counts partitions by number of distinct parts.
A326619/A326620 gives mean of distinct prime indices.
A326622 counts factorizations with integer mean, strict A328966.
A360071 counts partitions by number of parts and number of distinct parts.
The following count partitions:
- A360242 mean(parts) != mean(distinct parts), ranked by A360246.
- A360243 mean(parts) = mean(distinct parts), ranked by A360247.
- A360250 mean(parts) > mean(distinct parts), ranked by A360252.
- A360251 mean(parts) < mean(distinct parts), ranked by A360253.
Cf. A067539, A078174, A082550, A240219, A316313, A325347, A326567/A326568, A326669, A327475, A349156.

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

A326645 Heinz numbers of integer partitions whose mean and geometric mean are both integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 46, 47, 49, 53, 57, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193

Offset: 1

Gus Wiseman, Jul 16 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A326641.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}
   41: {13}
   43: {14}
   46: {1,9}
   47: {15}
   49: {4,4}

Wikipedia, Geometric mean

Heinz numbers of partitions with integer mean are A316413.
Heinz numbers of partitions with integer geometric mean are A326623.
Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.
Partitions with integer mean and geometric mean are A326641.
Subsets with integer mean and geometric mean are A326643.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A051293, A056239, A067538, A067539, A078175, A112798, A316413, A326623, A326644, A326646, A326647.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A326623 Heinz numbers of integer partitions whose geometric mean is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A326625 Number of strict integer partitions of n whose geometric mean is an integer.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A326641 Number of integer partitions of n whose mean and geometric mean are both integers.

Views