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A328026 Number of divisible pairs of consecutive divisors of n.

%I A328026 #24 Feb 23 2023 15:30:20
%S A328026 0,1,1,2,1,2,1,3,2,2,1,2,1,2,2,4,1,3,1,4,2,2,1,2,2,2,3,4,1,2,1,5,2,2,
%T A328026 2,2,1,2,2,4,1,4,1,4,2,2,1,2,2,3,2,4,1,4,2,4,2,2,1,2,1,2,2,6,2,4,1,4,
%U A328026 2,2,1,2,1,2,3,4,2,4,1,4,4,2,1,2,2,2,2,6,1,2,2,4,2,2,2,2,1,3,4,6,1,4,1,6,2
%N A328026 Number of divisible pairs of consecutive divisors of n.
%C A328026 The number m = 2^n, n >= 0, is the smallest for which a(m) = n. - _Marius A. Burtea_, Nov 20 2019
%H A328026 Antti Karttunen, <a href="/A328026/b328026.txt">Table of n, a(n) for n = 1..100000</a>
%F A328026 a(p^k) = k for any prime number p and k >= 0. - _Rémy Sigrist_, Oct 05 2019
%e A328026 The divisors of 500 are {1,2,4,5,10,20,25,50,100,125,250,500}, with consecutive divisible pairs {1,2}, {2,4}, {5,10}, {10,20}, {25,50}, {50,100}, {125,250}, {250,500}, so a(500) = 8.
%t A328026 Table[Length[Split[Divisors[n],!Divisible[#2,#1]&]]-1,{n,100}]
%o A328026 (PARI) a(n) = {my(d=divisors(n), nb=0); for (i=2, #d, if ((d[i] % d[i-1]) == 0, nb++)); nb;} \\ _Michel Marcus_, Oct 05 2019
%o A328026 (Magma) f:=func<n,i|D[i+1] mod D[i] eq 0 where D is Divisors(n) >;  g:=func<k| #[i:i in [1..#Divisors(k)-1]| f(k,i)]>; [g(n):n in [1..100]]; // _Marius A. Burtea_, Nov 20 2019
%Y A328026 Positions of 1's are A000040.
%Y A328026 Positions of 0's and 2's are A328028.
%Y A328026 Positions of terms > 2 are A328189.
%Y A328026 Successive pairs of consecutive divisors are counted by A129308.
%Y A328026 Cf. A000005, A027750, A033676, A060680, A060681, A060682, A060683, A193829, A328023, A328024, A328194.
%K A328026 nonn
%O A328026 1,4
%A A328026 _Gus Wiseman_, Oct 03 2019
%E A328026 Data section extended up to a(105) by _Antti Karttunen_, Feb 23 2023