A326625 -id:A326625 - cp's OEIS frontend

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

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 *)

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}]

A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 8, 3, 8, 5, 7, 8, 8, 7, 9, 8, 17, 11, 11, 8, 16, 17, 17, 14, 18, 17, 26, 19, 24, 20, 30, 28, 32, 27, 37, 35, 48, 37, 45, 37, 51, 51, 58, 50, 64, 62, 83, 73, 84, 69, 91, 89, 101, 97, 116, 111, 136, 123, 142, 138, 160, 161, 181, 171, 205, 199, 231, 221

Offset: 1

Naohiro Nomoto, Jan 27 2002

a(n) is the number of integer partitions of n whose geometric mean is an integer. - Gus Wiseman, Jul 19 2019

			From _Gus Wiseman_, Jul 19 2019: (Start)
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (41)     (33)      (421)      (44)
                    (1111)  (11111)  (222)     (1111111)  (2222)
                                     (111111)             (11111111)
(End)

Wikipedia, Geometric mean

Partitions with integer average are A067538.
Subsets whose geometric mean is an integer are A326027.
The Heinz numbers of these partitions are A326623.
The strict case is A326625.
Cf. A000041, A102627, A320322, A326028, A326641.

Table[Length[Select[IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,30}] (* Gus Wiseman, Jul 19 2019 *)

from math import prod
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A067539(n): return sum(1 for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) # Chai Wah Wu, Sep 24 2023

Terms a(61) onwards from Max Alekseyev, Feb 06 2010

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

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[#]]]&]

A326673 The positions of ones in the reversed binary expansion of n have integer geometric mean.

Original entry on oeis.org

1, 2, 4, 8, 9, 11, 16, 32, 64, 128, 130, 138, 256, 257, 261, 264, 296, 388, 420, 512, 1024, 2048, 2052, 2084, 2306, 2316, 2338, 2348, 4096, 8192, 16384, 32768, 32769, 32776, 32777, 32899, 32904, 32907, 33024, 35072, 65536, 131072, 131074, 131084, 131106

Offset: 1

Gus Wiseman, Jul 17 2019

			The reversed binary expansion of 11 is (1,1,0,1) and {1,2,4} has integer geometric mean, so 11 is in the sequence.

Andrew Howroyd, Table of n, a(n) for n = 1..211
Wikipedia, Geometric mean

Partitions with integer geometric mean are A067539.
Subsets with integer geometric mean are A326027.
Factorizations with integer geometric mean are A326028.
Numbers whose binary digit positions have integer mean are A326669.
Numbers whose binary digit positions are relatively prime are A326674.
Numbers whose binary digit positions have integer geometric mean are A326672.
Cf. A000120, A051293, A070939, A291166, A326625.

Select[Range[1000],IntegerQ[GeometricMean[Join@@Position[Reverse[IntegerDigits[#,2]],1]]]&]

ok(n)={ispower(prod(i=0, logint(n,2), if(bittest(n,i), i+1, 1)), hammingweight(n))}
{ for(n=1, 10^7, if(ok(n), print1(n, ", "))) } \\ Andrew Howroyd, Sep 29 2019

A326643 Number of subsets of {1..n} whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 30, 31, 32, 33, 34, 35, 41, 46, 47, 70, 71, 72, 73, 74, 102, 103, 104, 105, 143, 144, 145, 146, 151, 152, 153, 154, 155, 161, 162, 163, 244, 252, 280, 281, 282, 283, 409, 410, 416, 417, 418, 419

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(1) = 1 through a(12) = 16 subsets:
  {1}  {1}  {1}  {1}  {1}  {1}  {1}  {1}    {1}    {1}    {1}    {1}
       {2}  {2}  {2}  {2}  {2}  {2}  {2}    {2}    {2}    {2}    {2}
            {3}  {3}  {3}  {3}  {3}  {3}    {3}    {3}    {3}    {3}
                 {4}  {4}  {4}  {4}  {4}    {4}    {4}    {4}    {4}
                      {5}  {5}  {5}  {5}    {5}    {5}    {5}    {5}
                           {6}  {6}  {6}    {6}    {6}    {6}    {6}
                                {7}  {7}    {7}    {7}    {7}    {7}
                                     {8}    {8}    {8}    {8}    {8}
                                     {2,8}  {9}    {9}    {9}    {9}
                                            {1,9}  {10}   {10}   {10}
                                            {2,8}  {1,9}  {11}   {11}
                                                   {2,8}  {1,9}  {12}
                                                          {2,8}  {1,9}
                                                                 {2,8}
                                                                 {3,6,12}
                                                                 {3,4,9,12}

David Wasserman, Table of n, a(n) for n = 0..119
Wikipedia, Geometric mean

Partial sums of A326644.
Subsets whose geometric mean is an integer are A326027.
Subsets whose mean is an integer are A051293.
Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A067538, A067539, A078175, A082553, A102627, A326623, A326625, A326645, A326646, A326647.

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

More terms from David Wasserman, Aug 03 2019

A326850 Number of strict integer partitions of n whose maximum part divides n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 10, 1, 10, 5, 12, 1, 23, 1, 18, 15, 23, 1, 49, 1, 34, 36, 38, 1, 106, 1, 54, 79, 81, 1, 189, 1, 124, 162, 104, 1, 412, 1, 145, 307, 289, 1, 608, 12, 437, 559, 256, 1, 1432, 1, 340, 981, 976, 79, 1730, 1

Offset: 0

Gus Wiseman, Jul 28 2019

nonn

			The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5)
   6: (6)
   6: (3,2,1)
   7: (7)
   8: (8)
   8: (4,3,1)
   9: (9)
  10: (10)
  10: (5,4,1)
  10: (5,3,2)
  11: (11)
  12: (12)
  12: (6,5,1)
  12: (6,4,2)
  12: (6,3,2,1)
  13: (13)
  14: (14)
  14: (7,6,1)
  14: (7,5,2)
  14: (7,4,3)
  14: (7,4,2,1)
  15: (15)
  15: (5,4,3,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

Positions of 1's appear to be A308168.
The non-strict case is given by A067538.
Cf. A018818, A033630, A067538, A102627, A200745, A316413, A326625, A326836, A326843, A326851.

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

cp's OEIS Frontend

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

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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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A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

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

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

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

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A326673 The positions of ones in the reversed binary expansion of n have integer geometric mean.

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