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

Offset: 2

Author

Steven Lu, Jul 18 2023

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

For values at primes, see A364332.

Programs

Mathematica

Nest[Function[list,
  Module[{n = Length[list] + 1},
   Append[list,
    If[PrimeQ[n], Last[list] + 1,
     Max[(list[[First[#]]]) & /@ FactorInteger[n]]]]]], {0, 0}, 110]//Rest

Formula

a(2) = 0,

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

a(n) = max{a(k) : k|n, 1

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