This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362611 #19 Mar 02 2025 07:59:52
%S A362611 0,1,1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,2,1,1,1,2,1,1,1,3,1,1,2,2,
%T A362611 2,2,1,2,2,1,1,3,1,1,1,2,1,1,1,1,2,1,1,1,2,1,2,2,1,1,1,2,1,1,2,3,1,1,
%U A362611 2,3,1,1,1,2,1,1,2,3,1,1,1,2,1,1,2,2,2
%N A362611 Number of modes in the prime factorization of n.
%C A362611 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
%C A362611 a(n) depends only on the prime signature of n. - _Andrew Howroyd_, May 08 2023
%H A362611 Andrew Howroyd, <a href="/A362611/b362611.txt">Table of n, a(n) for n = 1..10000</a>
%F A362611 For n > 1, 1 <= a(n) << log n. - _Charles R Greathouse IV_, May 09 2023
%F A362611 a(n) <= A001221(n), with equality if and only if n is a power of a squarefree number (A072774). - _Amiram Eldar_, Mar 02 2025
%e A362611 The factorization of 450 is 2*3*3*5*5, modes {3,5}, so a(450) = 2.
%e A362611 The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so a(900) = 3.
%e A362611 The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so a(1500) = 1.
%e A362611 The factorization of 8820 is 2*2*3*3*5*7*7, modes {2,3,7}, so a(8820) = 3.
%t A362611 Table[x=Last/@If[n==1,0,FactorInteger[n]];Count[x,Max@@x],{n,100}]
%o A362611 (Python)
%o A362611 from sympy import factorint
%o A362611 def A362611(n): return list(v:=factorint(n).values()).count(max(v,default=0)) # _Chai Wah Wu_, May 08 2023
%o A362611 (PARI) a(n) = if(n==1, 0, my(f=factor(n)[,2], m=vecmax(f)); #select(v->v==m, f)) \\ _Andrew Howroyd_, May 08 2023
%Y A362611 Positions of first appearances are A002110.
%Y A362611 Positions of 1's are A356862, counted by A362608.
%Y A362611 Positions of terms > 1 are A362605, counted by A362607.
%Y A362611 For co-mode we have A362613, counted by A362615.
%Y A362611 This statistic (mode-count) has triangular form A362614.
%Y A362611 A027746 lists prime factors (with multiplicity).
%Y A362611 A112798 lists prime indices, length A001222, sum A056239.
%Y A362611 A359178 ranks partitions with a unique co-mode, counted by A362610.
%Y A362611 A362606 ranks partitions with more than one co-mode, counted by A362609.
%Y A362611 Cf. A000040, A000720, A001221, A002865, A072774, A215366, A327473, A327476, A359908.
%K A362611 nonn
%O A362611 1,6
%A A362611 _Gus Wiseman_, May 05 2023