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

Author

Gus Wiseman, Oct 03 2019

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

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.

Programs

Magma

f:=func;  g:=func; [g(n):n in [1..100]]; // Marius A. Burtea, Nov 20 2019

Mathematica

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

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

Formula

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

Extensions

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