A362047 -id:A362047 - cp's OEIS frontend

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.

Positions of zeros are 1 and A000040.
For length instead of mean we have A237832.
For minimum instead of mean we have A118096.
These partitions have ranks A362047.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean.
A097364 counts partitions by (maximum) - (minimum).
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]

A362268 Numbers whose prime factors counted with multiplicity satisfy: (maximum) - (minimum) = (mean).

Original entry on oeis.org

20, 60, 180, 189, 400, 540, 1200, 1372, 1620, 2541, 2835, 3185, 3600, 4860, 5577, 6860, 8000, 10800, 14365, 14580, 16093, 23465, 24000, 28812, 32400, 34300, 34375, 35721, 40733, 42525, 43740, 46529, 72000, 78793, 97200, 123101, 131220, 135401, 139755, 144060

Offset: 1

Chai Wah Wu, Apr 13 2023

nonn

			The terms together with their prime factors begin:
    20: [2, 2, 5]
    60: [2, 2, 3, 5]
   180: [2, 2, 3, 3, 5]
   189: [3, 3, 3, 7]
   400: [2, 2, 2, 2, 5, 5]
   540: [2, 2, 3, 3, 3, 5]
  1200: [2, 2, 2, 2, 3, 5, 5]
  1372: [2, 2, 7, 7, 7]
  1620: [2, 2, 3, 3, 3, 3, 5]
  2541: [3, 7, 11, 11]
  2835: [3, 3, 3, 3, 5, 7]
  3185: [5, 7, 7, 13]
  3600: [2, 2, 2, 2, 3, 3, 5, 5]
  4860: [2, 2, 3, 3, 3, 3, 3, 5]
The prime factors of 4860 are [2, 2, 3, 3, 3, 3, 3, 5], with minimum 2, maximum 5, and mean 3, and 5-2 = 3, so 4860 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A362047.

mmmQ[n_]:=With[{pf=Flatten[PadRight[{},#[[2]],#[[1]]]&/@FactorInteger[n]]},Max[pf]-Min[pf]==Mean[pf]]; Select[Range[150000],mmmQ] (* Harvey P. Dale, Aug 13 2025 *)

from itertools import count, islice
from math import prod
from sympy import factorint
def A362268_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(max(f:=factorint(n))-min(f))*sum(f.values())==sum(map(prod,f.items())),count(max(startvalue,2)))
A362268_list = list(islice(A362268_gen(),20))

cp's OEIS Frontend

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

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