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

0,29

Comments

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

Links

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

Wikipedia, Geometric mean

Example

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

Mathematica

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

Prog

(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

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.

Sequence in context: A319227 A219490 A341021 * A277142 A240592 A001877

Adjacent sequences: A358328 A358329 A358330 * A358332 A358333 A358334

Keyword

nonn

Author

Gus Wiseman, Nov 09 2022

Extensions

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

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

Status

approved