A357710 -id:A357710 - cp's OEIS frontend

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

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 *)

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

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

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