A102627 -id:A102627 - cp's OEIS frontend

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

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

A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

Original entry on oeis.org

1, 0, 3, 2, 12, 18, 101, 232, 1547, 3768, 24974, 116728, 687419, 3489664, 26436217, 159031250, 1129056772

Offset: 1

Gus Wiseman, Apr 04 2023

			The set partition y = {{1,4},{2,5},{3}} has block-means {5/2,7/2,3}, with mean 3, so y is counted under a(5).
The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}  {{12345}}
            {{13}{2}}    {{123}{4}}  {{1245}{3}}
            {{1}{2}{3}}              {{135}{24}}
                                     {{15}{234}}
                                     {{1}{234}{5}}
                                     {{12}{3}{45}}
                                     {{135}{2}{4}}
                                     {{14}{25}{3}}
                                     {{15}{24}{3}}
                                     {{1}{24}{3}{5}}
                                     {{15}{2}{3}{4}}
                                     {{1}{2}{3}{4}{5}}

For median instead of mean we have A361864.
For sum instead of outer mean we have A361866, median A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions whose block-sizes have integer mean.
A327475 counts subsets with integer mean, median A000975.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513, A326521, A326537, A327481.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Mean[Mean/@#]]&]],{n,6}]

a(13)-a(17) from Christian Sievers, Jun 30 2025

A363745 Number of integer partitions of n whose rounded-down mean is 2.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 4, 10, 6, 16, 21, 24, 32, 58, 47, 85, 111, 119, 158, 248, 217, 341, 442, 461, 596, 867, 792, 1151, 1465, 1506, 1916, 2652, 2477, 3423, 4298, 4381, 5488, 7334, 6956, 9280, 11503, 11663, 14429, 18781, 17992, 23383, 28675, 28970, 35449, 45203

Offset: 0

Gus Wiseman, Jul 05 2023

nonn

			The a(2) = 1 through a(10) = 16 partitions:
  (2)  .  (22)  (32)  (222)  (322)  (332)   (3222)  (3322)
          (31)  (41)  (321)  (331)  (422)   (3321)  (3331)
                      (411)  (421)  (431)   (4221)  (4222)
                             (511)  (521)   (4311)  (4321)
                                    (611)   (5211)  (4411)
                                    (2222)  (6111)  (5221)
                                    (3221)          (5311)
                                    (3311)          (6211)
                                    (4211)          (7111)
                                    (5111)          (22222)
                                                    (32221)
                                                    (33211)
                                                    (42211)
                                                    (43111)
                                                    (52111)
                                                    (61111)

For 1 instead of 2 we have A025065, ranks A363949.
The high version is A026905 reduplicated, ranks A363950.
Column k = 2 of A363945.
These partitions have ranks A363954.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A027336, A237984, A241131, A327472, A327482, A363723, A363943, A363944, A363946.

Table[Length[Select[IntegerPartitions[n],Floor[Mean[#]]==2&]],{n,0,30}]

A326670 Number of strict integer partitions y of n such that the average of the set {2^(s - 1): s in y} is an integer.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 6, 8, 7, 10, 9, 13, 12, 15, 16, 23, 22, 27, 31, 41, 41, 50, 57, 74, 75, 90, 99, 133, 127, 158, 167, 226, 203, 278, 262, 371, 325, 457, 387, 622, 484, 715, 606, 969, 672, 1178, 866, 1428, 1050, 1776, 1142, 2276, 1459, 2514, 1792

Offset: 1

Gus Wiseman, Jul 17 2019

nonn

			The a(1) = 1 through a(12) = 6 partitions (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)  (5)   (6)   (7)   (8)   (9)    (A)   (B)    (C)
                      (32)  (42)  (43)  (53)  (54)   (64)  (65)   (75)
                                  (52)  (62)  (63)   (73)  (74)   (84)
                                              (72)   (82)  (83)   (93)
                                              (531)        (92)   (A2)
                                                           (731)  (642)

The non-strict case is A326671.
Strict factorizations with integer average are A326668.
Strict partitions with integer average are A102627.
Cf. A001055, A051293, A067538, A102627, A326028, A326622, A326647, A326666, A326667.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[Mean[2^(#-1)]]&]],{n,30}]

A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 5, 0, 6, 3, 5, 0, 11, 6, 8, 7, 10, 0, 36, 0, 14, 16, 16, 29, 43, 0, 21, 36, 69, 0, 97, 0, 35, 138, 33, 0, 150, 61, 137, 134, 74, 0, 231, 134, 265, 229, 56, 0, 650, 0, 65, 749, 267, 247, 533, 0, 405, 565

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also strict partitions satisfying (maximum) = 2*(mean).

These are strict partitions where both the diagram and its complement (see example) have size n.

			The a(n) strict partitions for selected n (A..E = 10..14):
  n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
--------------------------------------------------------------
  621   831    7421   A32    8431   C42    A532   E43    B542
        6321          A41    8521   C51    A541   E52    B632
                                    9432   A631   E61    B641
                                    9531   A721          B731
                                    9621   85421         B821
                                           86321
The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
The strict partition y = (8,5,4,2,1) has diagram:
  o o o o o o o o
  o o o o o . . .
  o o o o . . . .
  o o . . . . . .
  o . . . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(20).

For minimum instead of mean we have A241035, non-strict A118096.
For length instead of mean we have A241087, non-strict A237753.
For median instead of mean we have A361850, non-strict A361849.
The non-strict version is A361853.
These partitions have ranks A361855 /\ A005117.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A008289 counts strict partitions by length.
A102627 counts strict partitions with integer mean, non-strict A067538.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[#]*Max@@#==2n&]],{n,30}]

A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

Original entry on oeis.org

1, 21, 57, 115, 133, 145, 159, 371, 393, 399, 515, 535, 565, 667, 803, 869, 917, 933, 1007, 1067, 1113, 1963, 2021, 2095, 2157, 2165, 2177, 2249, 2285, 2315, 2363, 2369, 2461, 2489, 2599, 2705, 2751, 2839, 2987, 3021, 3103, 3277, 3335, 3707, 3859, 4331, 4367

Offset: 1

Gus Wiseman, Sep 30 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
     1: {}
    21: {2,4}
    57: {2,8}
   115: {3,9}
   133: {4,8}
   145: {3,10}
   159: {2,16}
   371: {4,16}
   393: {2,32}
   399: {2,4,8}
   515: {3,27}
   535: {3,28}
   565: {3,30}
   667: {9,10}
   803: {5,21}
   869: {5,22}
   917: {4,32}
   933: {2,64}
  1007: {8,16}
  1067: {5,25}

The version including primes and nonsquarefree numbers is A326536.
The version for number of prime indices is A327900.
The version for sum of prime indices is A327901.
Cf. A001222, A005117, A056239, A078175, A102627, A112798, A316413, A326567/A326568, A326621, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Mean/@primeMS/@primeMS[#]&];

A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 1, 1, 5, 1, 1, 9, 7, 3, 1, 3, 1, 7, 11, 13, 1, 7, 1, 11, 35, 25, 31, 27, 5, 157, 1, 31, 131, 39, 31, 33, 37, 183, 179, 135, 157, 7, 265, 3, 871, 187, 865, 259, 879, 867, 179, 1593, 6073, 1593, 271, 5995, 149, 6661, 2411, 1509, 997, 1045, 5887

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(10) = 5 because we have [10], [9, 1], [1, 9], [8, 2] and [2, 8].

Eric W. Weisstein's World of Mathematics, Geometric Mean
Index entries for sequences related to compositions

For partitions we have A326625, non-strict A067539 (ranked by A326623).
The version for subsets is A326027.
For arithmetic mean we have A339175, non-strict A271654.
The non-strict case is counted by A357710, ranked by A357490.
A032020 counts strict compositions.
A067538 counts partitions with integer average.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A096199, A102627, A326622, A326624, A326028, A326641, A326645, A335405.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,15}] (* Gus Wiseman, Oct 30 2022 *)

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 3, 0, 4, 3, 4, 0, 8, 0, 10, 0, 11, 12, 19, 0, 0, 22, 0, 0, 46, 23, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of zeros are 2 and A013929.
The non-strict version is A330950 (A324851) q.v.
A000009 counts strict partitions.
A003963 multiplies together prime indices.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A057568 counts partitions whose product is divisible by their sum (A326149).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A112798 lists the prime indices of each positive integer.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A324925 counts partitions whose Heinz number is divisible by their product.
A326842 counts partitions whose parts and length all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A330952 counts partitions whose Heinz number is divisible by all parts.
A340828 counts strict partitions with length divisible by maximum.
A340830 counts strict partitions with parts divisible by length.
Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]

A361655 Number of even-length integer partitions of 2n with integer mean.

Original entry on oeis.org

0, 1, 3, 4, 10, 6, 33, 8, 65, 68, 117, 12, 583, 14, 319, 1078, 1416, 18, 3341, 20, 8035, 5799, 1657, 24, 36708, 16954, 3496, 24553, 68528, 30, 192180, 32, 178802, 91561, 14625, 485598, 955142, 38, 29223, 316085, 2622697, 42, 3528870, 44, 2443527, 5740043

Offset: 0

Gus Wiseman, Mar 23 2023

nonn

			The a(0) = 0 through a(5) = 6 partitions:
  .  (11)  (22)    (33)      (44)        (55)
           (31)    (42)      (53)        (64)
           (1111)  (51)      (62)        (73)
                   (111111)  (71)        (82)
                             (2222)      (91)
                             (3221)      (1111111111)
                             (3311)
                             (4211)
                             (5111)
                             (11111111)
For example, the partition (4,2,1,1) has length 4 and mean 2, so is counted under a(4).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Even-length partitions are counted by A027187, bisection A236913.
Including odd-length partitions gives A067538 bisected, ranks A316413.
For median instead of mean we have A361653.
The odd-length version is counted by A361656.
A000041 counts integer partitions, strict A000009.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A326622 counts factorizations with integer mean.
Cf. A000016, A027193, A067659, A082550, A102627, A237984, A240219, A327475, A348551, A359893.

Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,15}]

a(n)=if(n==0, 0, sumdiv(n, d, polcoef(1/prod(k=1, 2*d, 1 - x^k + O(x*x^(2*(n-d)))), 2*(n-d)))) \\ Andrew Howroyd, Mar 24 2023

Terms a(36) and beyond from Andrew Howroyd, Mar 24 2023

A326642 Number of non-constant integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 4, 2, 0, 0, 3, 1, 2, 1, 2, 0, 7, 0, 4, 2, 2, 4, 7, 0, 0, 4, 12, 0, 9, 0, 2, 11, 0, 0, 17, 6, 14, 4, 8, 0, 13, 6, 27, 6, 2, 0, 36, 0, 0, 35, 32, 8, 20, 0, 11, 6, 56, 0, 91, 0, 2, 17

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

The Heinz numbers of these partitions are given by A326646.

			The a(30) = 7 partitions:
  (27,3)
  (24,6)
  (24,3,3)
  (16,8,2,2,2)
  (9,9,9,1,1,1)
  (8,8,8,2,2,2)
  (8,8,4,4,1,1,1,1,1,1)

Wikipedia, Geometric mean

Partitions with integer mean and geometric mean are A326641.
Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.
Non-constant partitions with integer geometric mean are A326624.
Cf. A051293, A067538, A067539, A102627, A316413, A326029, A326643, A326644, A326645, A326647.

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

a(n) = A326641(n) - A000005(n).

cp's OEIS Frontend

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

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A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

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A363745 Number of integer partitions of n whose rounded-down mean is 2.

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A326670 Number of strict integer partitions y of n such that the average of the set {2^(s - 1): s in y} is an integer.

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A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

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A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

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A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

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A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

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A361655 Number of even-length integer partitions of 2n with integer mean.

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