A326623 -id:A326623 - cp's OEIS frontend

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

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

A326624 Heinz numbers of non-constant integer partitions whose geometric mean is an integer.

Original entry on oeis.org

14, 42, 46, 57, 76, 106, 126, 161, 183, 185, 194, 196, 228, 230, 302, 371, 378, 393, 399, 412, 424, 454, 477, 515, 588, 622, 679, 684, 687, 722, 742, 781, 786, 838, 1057, 1064, 1077, 1082, 1115, 1134, 1150, 1157, 1159, 1219, 1244, 1272, 1322, 1563, 1589, 1654

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:
    14: {1,4}
    42: {1,2,4}
    46: {1,9}
    57: {2,8}
    76: {1,1,8}
   106: {1,16}
   126: {1,2,2,4}
   161: {4,9}
   183: {2,18}
   185: {3,12}
   194: {1,25}
   196: {1,1,4,4}
   228: {1,1,2,8}
   230: {1,3,9}
   302: {1,36}
   371: {4,16}
   378: {1,2,2,2,4}
   393: {2,32}
   399: {2,4,8}
   412: {1,1,27}

Wikipedia, Geometric mean

The case with prime powers is A326623.
Subsets whose geometric mean is an integer are A326027.
Cf. A001055, A067539, A078175, A102627, A316413, A326567/A326568, A326622, A326625.

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

A271654 a(n) = Sum_{k|n} binomial(n-1,k-1).

Original entry on oeis.org

1, 2, 2, 5, 2, 17, 2, 44, 30, 137, 2, 695, 2, 1731, 1094, 6907, 2, 30653, 2, 97244, 38952, 352739, 2, 1632933, 10628, 5200327, 1562602, 20357264, 2, 87716708, 2, 303174298, 64512738, 1166803145, 1391282, 4978661179, 2, 17672631939, 2707475853, 69150651910, 2, 286754260229, 2, 1053966829029, 115133177854, 4116715363847, 2, 16892899722499, 12271514, 63207357886437

Offset: 1

Franklin T. Adams-Watters, Apr 11 2016

nonn

Also the number of compositions of n whose length divides n, i.e., compositions with integer mean, ranked by A096199. - Gus Wiseman, Sep 28 2022

			From _Gus Wiseman_, Sep 28 2022: (Start)
The a(1) = 1 through a(6) = 17 compositions with integer mean:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,1,1)  (1,3)      (1,1,1,1,1)  (1,5)
                       (2,2)                   (2,4)
                       (3,1)                   (3,3)
                       (1,1,1,1)               (4,2)
                                               (5,1)
                                               (1,1,4)
                                               (1,2,3)
                                               (1,3,2)
                                               (1,4,1)
                                               (2,1,3)
                                               (2,2,2)
                                               (2,3,1)
                                               (3,1,2)
                                               (3,2,1)
                                               (4,1,1)
                                               (1,1,1,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..3329

Cf. A056045.
The version for nonempty subsets is A051293, geometric A326027.
The version for partitions is A067538, ranked by A316413, strict A102627.
These compositions are ranked by A096199.
The version for factorizations is A326622, geometric A326028.
A011782 counts compositions.
A067539 = partitions w integer geo mean, ranked by A326623, strict A326625.
A100346 counts compositions into divisors, partitions A018818.
Cf. A000041, A078174, A078175, A326567/A326568, A326624, A326641, A326836, A326837, A326843.

a:= n-> add(binomial(n-1, d-1), d=numtheory[divisors](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 03 2023

Table[Length[Join @@ Permutations/@Select[IntegerPartitions[n],IntegerQ[Mean[#]]&]],{n,15}] (* Gus Wiseman, Sep 28 2022 *)

PARI
```
a(n)=sumdiv(n,k,binomial(n-1,k-1))
```

A082553 Number of sets of distinct positive integers whose geometric mean is an integer, the largest integer of a set is n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 7, 1, 1, 1, 9, 1, 29, 1, 3, 1, 1, 1, 31, 15, 1, 87, 3, 1, 1, 1, 115, 1, 1, 1, 257, 1, 1, 1, 17, 1, 1, 1, 3, 21, 1, 1, 519, 23, 141, 1, 3, 1, 847, 1, 19, 1, 1, 1, 215, 1, 1, 27, 1557, 1, 1, 1, 3, 1, 1, 1, 2617, 1, 1, 3125, 3, 1, 1

Offset: 1

Naohiro Nomoto, May 03 2003

nonn

a(n) = 1 if and only if n is squarefree (i.e., if and only if n is in A005117). - Nathaniel Johnston, Apr 28 2011

If n has a prime divisor p > sqrt(n), then a(n) = a(n/p). - Max Alekseyev, Aug 27 2013

			a(4) = 3: the three sets are {4}, {1, 4}, {1, 2, 4}.

Jinyuan Wang, Table of n, a(n) for n = 1..134
Wikipedia, Geometric mean

Subsets whose mean is an integer are A051293.
Partitions whose geometric mean is an integer are A067539.
Partial sums are A326027.
Strict partitions whose geometric mean is an integer are A326625.
Cf. A005117, A102627, A316413, A326623.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&IntegerQ[GeometricMean[#]]&]],{n,15}] (* Gus Wiseman, Jul 19 2019 *)

{ A082553(n) = my(m,c=0); if(issquarefree(n),return(1)); m = Set(vector(n-1,i,i)); forprime(p=sqrtint(n)+1,n, m = setminus(m,vector(n\p,i,p*i)); if(Mod(n,p)==0, return(A082553(n\p)) ); ); forvec(v=vector(#m,i,[0,1]), c += ispower(n*factorback(m,v),1+vecsum(v)) ); c; } \\ Max Alekseyev, Aug 31 2013

from sympy import factorint, factorial
def make_product(p, n, k):
    '''
    Find all k-element subsets of {1, ..., n} whose product is p.
    Returns: list of lists
    '''
    if n**k < p:
        return []
    if k == 1:
        return [[p]]
    if p%n == 0:
        l = [s + [n] for s in make_product(p//n, n - 1, k - 1)]
    else:
        l = []
    return l + make_product(p, n - 1, k)
def integral_geometric_mean(n):
    '''
    Find all subsets of {1, ..., n} that contain n and whose
    geometric mean is an integer.
    '''
    f = factorial(n)
    l = [[n]]
    #Find product of distinct prime factors of n
    c = 1
    for p in factorint(n):
        c *= p
    #geometric mean must be a multiple of c
    for gm in range(c, n, c):
        k = 2
        while not (gm**k%n == 0):
            k += 1
        while gm**k <= f:
            l += [s + [n] for s in make_product(gm**k//n, n - 1, k - 1)]
            k += 1
    return l
def A082553(n):
    return len(integral_geometric_mean(n)) # David Wasserman, Aug 02 2019

a(24)-a(62) from Max Alekseyev, Aug 31 2013

a(63)-a(99) from David Wasserman, Aug 02 2019

A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 6, 1, 3, 1, 2, 1, 1, 1, 3, 1, 6, 1, 5, 1, 2, 2, 2, 4, 3, 1, 9, 1, 1, 3, 1, 1, 4, 1, 4, 2, 6, 1, 6, 1, 3, 7, 4, 2, 5, 1, 10, 1, 3, 1, 9, 3

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(55) = 2 through a(60) = 9 partitions:
  (55)           (56)         (57)        (58)    (59)  (60)
  (27,16,9,2,1)  (24,18,8,6)  (49,7,1)    (49,9)        (54,6)
                              (27,25,5)   (50,8)        (48,12)
                              (27,18,12)                (27,24,9)
                                                        (27,24,6,2,1)
                                                        (36,12,9,2,1)
                                                        (36,9,6,4,3,2)
                                                        (24,18,9,6,2,1)
                                                        (27,16,9,4,3,1)

Wikipedia, Geometric mean

Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean are A102627.
Strict partitions with integer geometric mean are A326625.
Non-constant partitions with integer mean and geometric mean are A326641.
Subsets with integer mean and geometric mean are A326643.
Heinz numbers of partitions with integer mean and geometric mean are A326645.
Cf. A051293, A067538, A067539, A078175, A082553, A316413, A326027, A326623, A326644, A326646, A326647.

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

More terms from Jinyuan Wang, Jun 26 2020

A326644 Number of subsets of {1..n} containing n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 6, 5, 1, 23, 1, 1, 1, 1, 28, 1, 1, 1, 38, 1, 1, 1, 5, 1, 1, 1, 1, 6, 1, 1, 81, 8, 28, 1, 1, 1, 126, 1, 6, 1, 1, 1, 37, 1, 1, 6, 208, 1, 1, 1, 1, 1, 1, 1, 351, 1, 1, 381, 1, 1, 1, 1, 159, 605, 1, 1, 9, 1, 1, 1, 2, 1, 1223, 1, 1, 1, 1, 1, 805, 1, 113, 2, 5021, 1, 1, 1, 2, 1, 1, 1, 2630, 1, 1, 1, 54, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(1) = 1 through a(12) = 3 subsets:
  {1}  {2}  {3}  {4}  {5}  {6}  {7}  {8}    {9}    {10}  {11}  {12}
                                     {2,8}  {1,9}              {3,6,12}
                                                               {3,4,9,12}
The a(18) = 7 subsets:
  {18}
  {2,18}
  {8,18}
  {1,8,9,18}
  {2,3,8,9,18}
  {6,12,16,18}
  {2,3,4,9,12,18}

Wikipedia, Geometric mean

First differences of A326643.
Subsets whose mean is an integer are A051293.
Subsets whose geometric mean is an integer are A326027.
Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A067538, A067539, A082550, A082553, A102627, A326028, A326623, A326625, A326645, A326647.

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

More terms from David Wasserman, Aug 03 2019

A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892

Offset: 0

Gus Wiseman, Jan 23 2021

nonn

The only strict partitions counted are (), (1), and (2,1).

Is there a simple generating function?

			The a(1) = 1 through a(9) = 7 partitions:
  1  11  21   22    311    2211    331      2222      333
         111  1111  2111   111111  2221     4211      4221
                    11111          4111     221111    51111
                                   211111   311111    222111
                                   1111111  11111111  321111
                                                      21111111
                                                      111111111

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428), with strict case A340830.
The case where length also divides n is A326842 (A326847).
The Heinz numbers of these partitions are A340606.
The version for factorizations is A340851, with reciprocal version A340853.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A067539 counts partitions with integer geometric mean (A326623).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A330950 = partitions of n whose Heinz number is divisible by n (A324851).
Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]

A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

Original entry on oeis.org

36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2744, 2916, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400, 6724

Offset: 1

Gus Wiseman, Jul 17 2019

nonn

			36 has two such factorizations: (3*12) and (4*9).
The sequence of terms together with their prime indices begins:
    36: {1,1,2,2}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   144: {1,1,1,1,2,2}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   324: {1,1,2,2,2,2}
   400: {1,1,1,1,3,3}
   484: {1,1,5,5}
   512: {1,1,1,1,1,1,1,1,1}
   576: {1,1,1,1,1,1,2,2}
   676: {1,1,6,6}
   784: {1,1,1,1,4,4}
   900: {1,1,2,2,3,3}
  1000: {1,1,1,3,3,3}
  1024: {1,1,1,1,1,1,1,1,1,1}
  1156: {1,1,7,7}
  1296: {1,1,1,1,2,2,2,2}
  1444: {1,1,8,8}

Wikipedia, Geometric mean

A subsequence of A001597.
Factorizations with integer mean are A326622.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A067538, A067539, A326027, A326516, A326623, A326641, A326643, A326645, A326647.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Length[Select[facs[#],!IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]]>1&]

A326646 Heinz numbers of non-constant integer partitions whose mean and geometric mean are both integers.

Original entry on oeis.org

46, 57, 183, 194, 228, 371, 393, 454, 515, 687, 742, 838, 1057, 1064, 1077, 1150, 1157, 1159, 1244, 1322, 1563, 1895, 2018, 2060, 2116, 2157, 2163, 2167, 2177, 2225, 2231, 2405, 2489, 2854, 2859, 3249, 3263, 3339, 3352, 3558, 3669, 3758, 3787, 3914, 4265, 4351

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 A326642.

			The sequence of terms together with their prime indices begins:
    46: {1,9}
    57: {2,8}
   183: {2,18}
   194: {1,25}
   228: {1,1,2,8}
   371: {4,16}
   393: {2,32}
   454: {1,49}
   515: {3,27}
   687: {2,50}
   742: {1,4,16}
   838: {1,81}
  1057: {4,36}
  1064: {1,1,1,4,8}
  1077: {2,72}
  1150: {1,3,3,9}
  1157: {6,24}
  1159: {8,18}
  1244: {1,1,64}
  1322: {1,121}

Wikipedia, Geometric mean

Heinz numbers of partitions with integer mean and geometric mean are A326645.
Heinz numbers of partitions with integer mean are A316413.
Heinz numbers of partitions with integer geometric mean are A326623.
Non-constant partitions with integer mean and geometric mean are A326642.
Subsets with integer mean and geometric mean are A326643.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A051293, A067538, A067539, A078175, A082553, A102627, A326027, A326641, A326644, A326647.

A357710 Number of integer compositions of n with integer geometric mean.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 8, 4, 15, 17, 22, 48, 40, 130, 88, 287, 323, 543, 1084, 1145, 2938, 3141, 6928, 9770, 15585, 29249, 37540, 78464, 103289, 194265, 299752, 475086, 846933, 1216749, 2261920, 3320935, 5795349, 9292376, 14825858, 25570823, 39030115, 68265801, 106030947, 178696496

Offset: 0

Gus Wiseman, Oct 15 2022

nonn

			The a(6) = 4 through a(9) = 15 compositions:
  (6)       (7)        (8)         (9)
  (33)      (124)      (44)        (333)
  (222)     (142)      (2222)      (1224)
  (111111)  (214)      (11111111)  (1242)
            (241)                  (1422)
            (412)                  (2124)
            (421)                  (2142)
            (1111111)              (2214)
                                   (2241)
                                   (2412)
                                   (2421)
                                   (4122)
                                   (4212)
                                   (4221)
                                   (111111111)

The unordered version (partitions) is A067539, ranked by A326623.
Compositions with integer average are A271654, partitions A067538.
Subsets whose geometric mean is an integer are A326027.
The version for factorizations is A326028.
The strict case is A339452, partitions A326625.
These compositions are ranked by A357490.
A011782 counts compositions.
Cf. A025047, A051293, A078174, A078175, A102627, A320322, A326622, A326624, A326641, A357182, A357183.

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,0,15}]

from math import prod, factorial
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A357710(n): return sum(factorial(s)//prod(factorial(d) for d in p.values()) for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) if n else 0 # Chai Wah Wu, Sep 24 2023

More terms from David A. Corneth, Oct 17 2022

cp's OEIS Frontend

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

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A326624 Heinz numbers of non-constant integer partitions whose geometric mean is an integer.

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A271654 a(n) = Sum_{k|n} binomial(n-1,k-1).

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A082553 Number of sets of distinct positive integers whose geometric mean is an integer, the largest integer of a set is n.

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A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

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A326644 Number of subsets of {1..n} containing n whose mean and geometric mean are both integers.

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A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

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Mathematica

A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

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