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

%I A362268 #12 Aug 13 2025 19:25:57
%S A362268 20,60,180,189,400,540,1200,1372,1620,2541,2835,3185,3600,4860,5577,
%T A362268 6860,8000,10800,14365,14580,16093,23465,24000,28812,32400,34300,
%U A362268 34375,35721,40733,42525,43740,46529,72000,78793,97200,123101,131220,135401,139755,144060
%N A362268 Numbers whose prime factors counted with multiplicity satisfy: (maximum) - (minimum) = (mean).
%H A362268 Harvey P. Dale, <a href="/A362268/b362268.txt">Table of n, a(n) for n = 1..100</a>
%e A362268 The terms together with their prime factors begin:
%e A362268     20: [2, 2, 5]
%e A362268     60: [2, 2, 3, 5]
%e A362268    180: [2, 2, 3, 3, 5]
%e A362268    189: [3, 3, 3, 7]
%e A362268    400: [2, 2, 2, 2, 5, 5]
%e A362268    540: [2, 2, 3, 3, 3, 5]
%e A362268   1200: [2, 2, 2, 2, 3, 5, 5]
%e A362268   1372: [2, 2, 7, 7, 7]
%e A362268   1620: [2, 2, 3, 3, 3, 3, 5]
%e A362268   2541: [3, 7, 11, 11]
%e A362268   2835: [3, 3, 3, 3, 5, 7]
%e A362268   3185: [5, 7, 7, 13]
%e A362268   3600: [2, 2, 2, 2, 3, 3, 5, 5]
%e A362268   4860: [2, 2, 3, 3, 3, 3, 3, 5]
%e A362268 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.
%t A362268 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 *)
%o A362268 (Python)
%o A362268 from itertools import count, islice
%o A362268 from math import prod
%o A362268 from sympy import factorint
%o A362268 def A362268_gen(startvalue=2): # generator of terms >= startvalue
%o A362268     return filter(lambda n:(max(f:=factorint(n))-min(f))*sum(f.values())==sum(map(prod,f.items())),count(max(startvalue,2)))
%o A362268 A362268_list = list(islice(A362268_gen(),20))
%Y A362268 Cf. A362047.
%K A362268 nonn
%O A362268 1,1
%A A362268 _Chai Wah Wu_, Apr 13 2023