A364334 - OEIS

A364334

a(2) = 0; a(n) = a(n-1) + 1 if n is an odd prime; otherwise a(n) = max{a(k) : k is divisor of n, 1 < k < n}.

0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 1, 0, 1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 0, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 1, 3, 4, 1, 2, 1, 1, 2, 3, 1, 2, 2, 2, 3, 4, 1, 2, 2, 2, 0, 2, 2, 3, 1, 3, 2, 3, 1, 2, 2, 1, 2, 2, 2, 3, 1, 1, 2, 3, 2, 1, 3, 3, 2, 3, 1, 2, 3, 2, 4, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 4, 1, 2, 2, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,6

COMMENTS

This sequence is a kind of measure of the "amount of information" in an integer. The post at Zhihu wonders whether one can calculate this sequence without using prime decomposition.

LINKS

Table of n, a(n) for n=2..112.

Zhihu, Can the order of a number be known by bypassing the complicated calculation of "prime factor decomposing"?, Aug 12 2022.

FORMULA

a(2) = 0,

a(n) = a(n-1) + 1 if n is an odd prime,

a(n) = max{a(k) : k|n, 1<k<n} otherwise.

EXAMPLE

a(238)=2, since a(2)=0, a(7)=2, a(14)=2, a(17)=1, a(34)=1, a(119)=2, and the largest among them is 2.

And a(239)=3, since 239 is a prime number, and a(238)=2.

MATHEMATICA

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