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

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

Author

Gus Wiseman, Nov 09 2022

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Python

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

Extensions

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

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