A326027 -id:A326027 - cp's OEIS frontend

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

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

A339454 Number of subsets of {1..n} whose root mean square is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 15, 20, 29, 52, 87, 166, 311, 538, 943, 1682, 2915, 5054, 8905, 15904, 28533, 51826, 95191, 175402, 325777, 607542, 1134191, 2128922, 3986433, 7485522, 14065135, 26446388, 49796025, 93920770, 177470237, 335780796, 636883269, 1209603646

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(9) = 15 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {1, 7}, {1, 5, 7}, {1, 3, 5, 8, 9}, {3, 4, 5, 7, 9}, {1, 3, 5, 6, 8, 9} and {3, 4, 5, 6, 7, 9}.

Max Alekseyev, Table of n, a(n) for n = 1..100
Eric W. Weisstein's World of Mathematics, Root-Mean-Square

Cf. A051293, A240089, A326027, A339453.

from functools import lru_cache
from sympy.ntheory.primetest import is_square
def cond(sos, c): return c > 0 and sos%c == 0 and is_square(sos//c)
@lru_cache(maxsize=None)
def b(n, sos, c):
    if n == 0: return int(cond(sos, c))
    return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Oct 06 2022

a(n) = A357415(n) + A357416(n). - Max Alekseyev, Mar 25 2025

a(23)-a(40) from Alois P. Heinz, Dec 05 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

A339453 Number of subsets of {1..n} whose harmonic mean is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 13, 14, 15, 18, 19, 26, 27, 30, 53, 54, 55, 100, 101, 180, 203, 210, 211, 378, 379, 382, 383, 1092, 1093, 2020, 2021, 2022, 3933, 3956, 6473, 10226, 10227, 10266, 10561, 20948, 20949

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

For terms listed in the Data section: a(p^k) = a(p^k-1) + 1, where p prime (empirical observation). - Ilya Gutkovskiy, Dec 06 2020
From Chai Wah Wu, Dec 14 2020: (Start)
The above empirical observation is true.
Theorem: For prime p, a(p^k) = a(p^k-1)+1.
Proof: Since the singleton set {x} has harmonic mean x, a(n) >= a(n-1)+1.
Let S = {s_1,s_2,..,s_n} be a subset of {1,2,..,p^k} with n>1 elements such that s_n = p^k and let H be the harmonic mean of S. Let M = A003418(p^k) be the least common multiple of {1,2,..,p^k}. Then M = Wp^k where p does not divide W = A002944(p^k).
Let Q_i = M/s_i and Q = sum_i Q_i. This implies that Q_n = W and p divides Q_i for i < n.
H can be written as nM/Q. Since p does not divide W, this implies that p does not divide Q. Suppose H is an integer. Then this implies that Q divides nM/p^k = nW.
Note that s_i < s_n for i < n. This implies that Q_i > W for i < n, i.e. Q > nW, and this contradicts the fact that Q divides nW and thus H is not an integer.
Thus {p^k} is the only subset of {1,2,..,p^k} that includes p^k and have an integral Harmonic mean.
This concludes the proof.
(End)

			a(6) = 12 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {2, 6}, {3, 6}, {1, 3, 6}, {2, 3, 6}, {3, 4, 6} and {1, 2, 3, 6}.

Eric W. Weisstein's World of Mathematics, Harmonic Mean

Cf. A002944, A003418, A051293, A067540, A246655, A326027, A339454.

from itertools import chain, combinations
from fractions import Fraction
def powerset(s): # skip empty set
    return chain.from_iterable(combinations(s, r) for r in range(1,len(s)+1))
def hm(s):
    ss = sum(Fraction(1, i) for i in s)
    return Fraction(len(s)*ss.denominator, ss.numerator)
def a(n):
    return sum(hm(s).denominator==1 for s in powerset(range(1,n+1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Dec 06 2020

from math import lcm
from itertools import combinations
def A339453(n):
    m = lcm(*range(2,n+1))
    return sum(1 for i in range(1,n+1) for d in combinations((m//i for i in range(1,n+1)),i) if m*i % sum(d) == 0) # Chai Wah Wu, Dec 02 2021

a(n) >= a(n-1)+1. For prime p, a(p^k) = a(p^k-1)+1. - Chai Wah Wu, Dec 14 2020

a(n) = A357411(n) + A357412(n). - Max Alekseyev, Feb 26 2025

a(23)-a(29) from Michael S. Branicky, Dec 06 2020
a(30)-a(35) from Chai Wah Wu, Dec 08 2020
a(36)-a(39) from Chai Wah Wu, Dec 11 2020
a(40)-a(41) from Chai Wah Wu, Dec 19 2020

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.

A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 19, 19, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 39, 39, 40, 40, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 90, 90, 91, 91, 92, 92

Offset: 0

Ilya Gutkovskiy, Sep 27 2022

nonn

The geometric mean of a subset such as in name must be an odd number in {1..n} which might ease the search for terms. - David A. Corneth, Sep 29 2022

			a(9) = 7 subsets: {1}, {3}, {5}, {7}, {9}, {1, 9} and {1, 3, 9}.

Max Alekseyev, Table of n, a(n) for n = 0..256

Cf. A001055, A326027, A357355, A357411, A357414, A357415.

from functools import lru_cache
from sympy import integer_nthroot
def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1
@lru_cache(maxsize=None)
def b(n, p, c):
    if n == 0: return int (c > 0 and cond(p, c))
    return b(n-1, p, c) + b(n-1, p*n, c+1) if n&1 else b(n-1, p, c)
@lru_cache(maxsize=None)
def a(n): return b(n, 1, 0) if n&1 else b(n-1, 1, 0) if n else 0
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 29 2022

a(2*n-1) = a(2*n) for n >= 1. - David A. Corneth, Sep 29 2022

a(n) = A326027(n) - A357414(n). - Max Alekseyev, Mar 01 2025

a(24)-a(34) from Michael S. Branicky, Sep 29 2022
a(35)-a(70) from David A. Corneth, Sep 29 2022
a(0) prepended and terms a(71) onward added by Max Alekseyev, Mar 06 2025

A357414 Number of nonempty subsets of {1..n} whose elements have an even geometric mean.

Original entry on oeis.org

0, 0, 1, 1, 4, 4, 5, 5, 8, 12, 13, 13, 20, 20, 21, 21, 30, 30, 59, 59, 62, 62, 63, 63, 94, 104, 105, 187, 190, 190, 191, 191, 306, 306, 307, 307, 564, 564, 565, 565, 582, 582, 583, 583, 586, 600, 601, 601, 1120, 1134, 1275, 1275, 1278, 1278, 2125, 2125, 2144, 2144, 2145, 2145, 2360, 2360, 2361, 2381, 3938, 3938, 3939, 3939, 3942, 3942, 3943, 3943, 6560, 6560, 6561, 9663, 9666

Offset: 0

Ilya Gutkovskiy, Sep 27 2022

nonn

			a(8) = 8 subsets: {2}, {4}, {6}, {8}, {1, 4}, {2, 8}, {1, 2, 4} and {2, 4, 8}.

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

Cf. A326027, A357356, A357412, A357413, A357416.

from functools import lru_cache
from sympy import integer_nthroot
def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1 == 0
@lru_cache(maxsize=None)
def b(n, p, c):
    if n == 0: return int (c > 0 and cond(p, c))
    return b(n-1, p, c) + b(n-1, p*n, c+1)
a = lambda n: b(n, 1, 0)
print([a(n) for n in range(26)]) # Michael S. Branicky, Sep 29 2022

a(p) = a(p-1) for prime p > 2. - Michael S. Branicky, Sep 30 2022

a(n) = A326027(n) - A357413(n). - Max Alekseyev, Mar 06 2025

a(24)-a(41) from Michael S. Branicky, Sep 30 2022
Terms a(42) onward from Max Alekseyev, Oct 11 2023
a(0) prepended by Max Alekseyev, Mar 06 2025

cp's OEIS Frontend

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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A339454 Number of subsets of {1..n} whose root mean square is an integer.

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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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A339453 Number of subsets of {1..n} whose harmonic mean is an integer.

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