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A356862 Numbers with a unique largest prime exponent.

%I A356862 #53 Sep 17 2024 04:02:44
%S A356862 2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,37,40,
%T A356862 41,43,44,45,47,48,49,50,52,53,54,56,59,60,61,63,64,67,68,71,72,73,75,
%U A356862 76,79,80,81,83,84,88,89,90,92,96,97,98,99,101,103,104
%N A356862 Numbers with a unique largest prime exponent.
%C A356862 If the prime factorization of k has a unique largest exponent, then k is a term.
%C A356862 Numbers whose multiset of prime factors (with multiplicity) has a unique mode. - _Gus Wiseman_, May 12 2023
%C A356862 Disjoint union of A246655 and A376250. The asymptotic density of this sequence, 0.3660366524547281232052..., is equal to the density of A376250 since the prime powers have a zero density. - _Amiram Eldar_, Sep 17 2024
%H A356862 Jens Ahlström, <a href="/A356862/b356862.txt">Table of n, a(n) for n = 1..10000</a>
%H A356862 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mode_(statistics)">Mode (statistics)</a>.
%e A356862 Prime powers (A246655) are in the sequence, since they have only one prime exponent in their prime factorization, hence a unique largest exponent.
%e A356862 144 is in the sequence, since 144 = 2^4 * 3^2 and there is the unique largest exponent 4.
%e A356862 225 is not in the sequence, since 225 = 3^2 * 5^2 and the largest exponent 2 is not unique, but rather it is the exponent of both the prime factor 3 and of the prime factor 5.
%t A356862 Select[Range[2, 100], Count[(e = FactorInteger[#][[;; , 2]]), Max[e]] == 1 &] (* _Amiram Eldar_, Sep 01 2022 *)
%o A356862 (Python)
%o A356862 from sympy import factorint
%o A356862 from collections import Counter
%o A356862 def ok(k):
%o A356862     c = Counter(factorint(k)).most_common(2)
%o A356862     return not (len(c) > 1 and c[0][1] == c[1][1])
%o A356862 print([k for k in range(2, 105) if ok(k)])
%o A356862 (Python)
%o A356862 from sympy import factorint
%o A356862 from itertools import count, islice
%o A356862 def A356862_gen(startvalue=2): # generator of terms >= startvalue
%o A356862     return filter(lambda n:len(f:=sorted(factorint(n).values(),reverse=True))==1 or f[0]!=f[1],count(max(startvalue,2)))
%o A356862 A356862_list = list(islice(A356862_gen(),30)) # _Chai Wah Wu_, Sep 10 2022
%o A356862 (PARI) isok(k) = if (k>1, my(f=factor(k), m=vecmax(f[,2]), w=select(x->(f[x,2] == m), [1..#f~])); #w == 1); \\ _Michel Marcus_, Sep 01 2022
%Y A356862 Subsequence of A319161 (which has additional terms 1, 180, 252, 300, 396, 450, 468, ...).
%Y A356862 Cf. A246655, A356838, A356840, A376250.
%Y A356862 For factors instead of exponents we have A102750.
%Y A356862 For smallest instead of largest we have A359178, counted by A362610.
%Y A356862 The complement is A362605, counted by A362607.
%Y A356862 The complement for co-mode is A362606, counted by A362609.
%Y A356862 Partitions of this type are counted by A362608.
%Y A356862 These are the positions of 1's in A362611, for co-modes A362613.
%Y A356862 A001221 is the number of prime exponents, sum A001222.
%Y A356862 A027746 lists prime factors, A112798 indices, A124010 exponents.
%Y A356862 A362614 counts partitions by number of modes, A362615 co-modes.
%Y A356862 Cf. A000041, A002865, A051903, A056239, A070003, A247180, A283050, A327473.
%K A356862 nonn,easy
%O A356862 1,1
%A A356862 _Jens Ahlström_, Sep 01 2022