A328026 Number of divisible pairs of consecutive divisors of n.

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, 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, 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

Offset

1,4

Comments

The number m = 2^n, n >= 0, is the smallest for which a(m) = n. - Marius A. Burtea, Nov 20 2019

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(p^k) = k for any prime number p and k >= 0. - Rémy Sigrist, Oct 05 2019

Example

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.

Mathematica

Table[Length[Split[Divisors[n], !Divisible[#2, #1]&]]-1, {n, 100}]

Prog

(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
(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

Crossrefs

Positions of 1's are A000040.

Positions of 0's and 2's are A328028.

Positions of terms > 2 are A328189.

Successive pairs of consecutive divisors are counted by A129308.

Cf. A000005, A027750, A033676, A060680, A060681, A060682, A060683, A193829, A328023, A328024, A328194.

Sequence in context: A076558 A328195 A235875 * A326975 A204893 A251717

Adjacent sequences: A328023 A328024 A328025 * A328027 A328028 A328029

Keyword

nonn

Author

Gus Wiseman, Oct 03 2019

Extensions

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

Status

approved