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

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

Author

Chai Wah Wu, Apr 13 2023

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A362047.

Programs

Mathematica

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 *)

Python

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))