A067539 -id:A067539 - cp's OEIS frontend

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

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

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 *)

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).

A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 15, 16, 17, 24, 31, 32, 36, 42, 63, 64, 69, 70, 81, 88, 98, 104, 127, 128, 136, 170, 255, 256, 277, 278, 282, 292, 325, 326, 337, 344, 354, 360, 394, 418, 424, 511, 512, 513, 514, 515, 528, 547, 561, 568, 640, 682, 768, 769, 785, 792, 896

Offset: 1

Gus Wiseman, Oct 16 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  24: (1,4)
  31: (1,1,1,1,1)
  32: (6)
  36: (3,3)
  42: (2,2,2)
  63: (1,1,1,1,1,1)
  64: (7)
  69: (4,2,1)

For regular mean we have A096199, counted by A271654 (partitions A067538).
Subsets whose geometric mean is an integer are counted by A326027.
The unordered version (partitions) is A326623, counted by A067539.
The strict case is counted by A339452, partitions A326625.
These compositions are counted by A357710.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A102627, A301987, A326622, A326624, A326028, A326567/A326568, A326641, A326645, A335405, A357184.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],IntegerQ[GeometricMean[stc[#]]]&]

A358331 Number of integer partitions of n with arithmetic and geometric mean differing by one.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 3, 3, 0, 0, 2, 2, 0, 4, 0, 0, 5, 0, 0, 4, 5, 4, 3, 2, 0, 3, 3, 10, 4, 0, 0, 7, 0, 0, 16, 2, 4, 4, 0, 0, 5, 24, 0, 6, 0, 0, 9, 0, 27, 10, 0, 7, 7, 1, 0, 44

Offset: 0

Gus Wiseman, Nov 09 2022

nonn

The arithmetic and geometric mean from such partition is a positive integer. - David A. Corneth, Nov 11 2022

			The a(30) = 2 through a(36) = 3 partitions (C = 12, G = 16):
  (888222)      .  (99333311)  (G2222222111)  .  (C9662)    (G884)
  (8844111111)                                   (C9833)    (888222111111)
                                                 (8884421)  (G42222221111)

Chai Wah Wu, Table of n, a(n) for n = 0..143
Wikipedia, Geometric mean

The version for subsets seems to be close to A178832.
These partitions are ranked by A358332.
A000041 counts partitions.
A067538 counts partitions with integer average, ranked by A316413.
A067539 counts partitions with integer geometric mean, ranked by A326623.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A078174, A102627, A271654, A326027, A326624, A326625, A326028, A326641, A326645, A357710.

Table[Length[Select[IntegerPartitions[n],Mean[#]==1+GeometricMean[#]&]],{n,0,30}]

a(n) = if (n, my(nb=0,vp); forpart(p=n, vp=Vec(p); if (vecsum(vp)/#p == 1 + sqrtn(vecprod(vp), #p), nb++)); nb, 0); \\ Michel Marcus, Nov 11 2022

from math import prod
from sympy import divisors, integer_nthroot
from sympy.utilities.iterables import partitions
def A358331(n):
    divs = {d:n//d-1 for d in divisors(n,generator=True)}
    return sum(1 for s,p in partitions(n,m=max(divs,default=0),size=True) if s in divs and (t:=integer_nthroot(prod(a**b for a, b in p.items()),s))[1] and divs[s]==t[0]) # Chai Wah Wu, Sep 24 2023

a(61)-a(80) from Giorgos Kalogeropoulos, Nov 11 2022

a(81)-a(84) from Chai Wah Wu, Sep 24 2023

A358332 Numbers whose prime indices have arithmetic and geometric mean differing by one.

Original entry on oeis.org

57, 228, 1064, 1150, 1159, 2405, 3249, 7991, 29785, 29999, 30153, 35378, 51984, 82211, 133931, 185193, 187039, 232471, 242592, 374599, 404225, 431457, 685207, 715129, 927288, 1132096, 1165519, 1322500, 1343281, 1555073, 1872413, 2016546, 2873687, 3468319, 4266421, 4327344

Offset: 1

Gus Wiseman, Nov 09 2022

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 terms together with their prime indices begin:
      57: {2,8}
     228: {1,1,2,8}
    1064: {1,1,1,4,8}
    1150: {1,3,3,9}
    1159: {8,18}
    2405: {3,6,12}
    3249: {2,2,8,8}
    7991: {18,32}
   29785: {3,4,9,12}
   29999: {32,50}
   30153: {2,8,9,9}
   35378: {1,4,4,8,8}
   51984: {1,1,1,1,2,2,8,8}
   82211: {50,72}
  133931: {4,8,8,16}
  185193: {2,2,2,8,8,8}
  187039: {72,98}
  232471: {12,18,27}

Max Alekseyev, Table of n, a(n) for n = 1..1458
Wikipedia, Geometric mean

The partitions with these Heinz numbers are counted by A358331.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A067538 counts partitions with integer average, ranked by A316413.
A067539 counts partitions with integer geometric mean, ranked by A326623.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A000720, A051293, A111233, A215366, A289508, A289509, A326027, A326624, A326028, A326645, A357710.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],Mean[primeMS[#]]==1+GeometricMean[primeMS[#]]&]

isok(k) = if (k>1, my(f=factor(k), vf=List()); for (i=1, #f~, for (j=1, f[i,2], listput(vf, primepi(f[i,1])))); my(v = Vec(vf)); vecsum(v)/#v == 1 + sqrtn(vecprod(v), #v);); \\ Michel Marcus, Nov 11 2022

More terms from Michel Marcus, Nov 11 2022

cp's OEIS Frontend

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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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.

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

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

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A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

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A358331 Number of integer partitions of n with arithmetic and geometric mean differing by one.

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