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Maximum length of a divisibility chain of consecutive divisors of n greater than 1.
8

%I #15 Dec 07 2024 08:01:40

%S 0,1,1,2,1,2,1,3,2,2,1,2,1,2,2,4,1,2,1,3,2,2,1,2,2,2,3,3,1,2,1,5,2,2,

%T 2,2,1,2,2,3,1,2,1,3,2,2,1,2,2,2,2,3,1,2,2,3,2,2,1,2,1,2,2,6,2,2,1,3,

%U 2,2,1,2,1,2,2,3,2,2,1,3,4,2,1,2,2,2,2,4,1,2,2,3,2,2,2,2,1,2,3,3,1,2,1,4,2

%N Maximum length of a divisibility chain of consecutive divisors of n greater than 1.

%C Also the maximum length of a divisibility chain of consecutive divisors of n less than n.

%C The divisors of n (except 1) are row n of A027749.

%H Antti Karttunen, <a href="/A328195/b328195.txt">Table of n, a(n) for n = 1..65537</a>

%e The divisors of 272 greater than 1 are {2, 4, 8, 16, 17, 34, 68, 136, 272}, with divisibility chains {{2, 4, 8, 16}, {17, 34, 68, 136, 272}}, so a(272) = 5.

%t Table[If[n==1,0,Max@@Length/@Split[DeleteCases[Divisors[n],1],Divisible[#2,#1]&]],{n,100}]

%o (PARI) A328195(n) = if(1==n, 0, my(divs=divisors(n), rl=0,ml=1); for(i=2,#divs,if(!(divs[i]%divs[i-1]), rl++, ml = max(rl,ml); rl=1)); max(ml,rl)); \\ _Antti Karttunen_, Dec 07 2024

%Y Allowing 1 as a divisor gives A328162.

%Y Forbidding n as a divisor of n gives A328194.

%Y Positions of 1's are A000040 (primes).

%Y Indices of terms greater than 1 are A002808 (composite numbers).

%Y The maximum run-length of divisors of n is A055874(n).

%Y Cf. A000005, A033676, A060775, A163870, A328026, A328161, A328171.

%K nonn

%O 1,4

%A _Gus Wiseman_, Oct 14 2019

%E Data section extended up to a(105) by _Antti Karttunen_, Dec 07 2024