A061373 "Natural" logarithm, defined inductively by a(1)=1, a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if n, m>1.

1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 10, 11, 9, 10, 10, 9, 10, 11, 10, 11, 10, 11, 11, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 11, 13, 14, 11, 12, 12, 12, 12, 13, 11, 13, 12, 12, 13, 14, 12, 13, 13, 12, 12, 13, 13, 14, 13, 14, 13, 14, 12, 13, 13, 13, 13, 14

Offset: 1

Author

Juan Arias-de-Reyna, Jun 08 2001

Keywords

• easy
• nice
• nonn

Comments

Related to A005245, the complexity of n, which is <= this sequence. They are equal up to term a(46) and for 771 values out of the first 1000 terms. A061373 is easier to compute.

a(A182061(n)) = n and a(m) < n for m < A182061(n). [Reinhard Zumkeller, Apr 09 2012]

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• J. Arias de Reyna, Complejidad de los numeros naturales, Gaceta de la Real Sociedad Matematica Espanola, 3, (2000), 230-250. (In Spanish.)
• J. Arias de Reyna, Complejidad de los numeros naturales, Gaceta de la Real Sociedad Matematica Espanola, 3, (2000), 230-250. (In Spanish.) [Cached copy, with permission]
• J. Arias de Reyna, Complexity of natural numbers, arXiv:2111.03345 [math.NT], 2021.

Crossrefs

Cf. A005245.

Cf. A020639.

Programs

Haskell

import Data.List (genericIndex)
a061373 1 = 1
a061373 n = genericIndex a061373_list (n-1)
a061373_list = 1 : f 2 where
   f x | x == spf  = 1 + a061373 (spf - 1) : f (x + 1)
       | otherwise = a061373 spf + a061373 (x `div` spf) : f (x + 1)
       where spf = a020639 x
-- Reinhard Zumkeller, Apr 09 2012

Mathematica

a[1]=1; a[p_?PrimeQ] := 1+a[p-1]; a[n_] := a[n] = With[{d=Divisors[n][[2]] }, a[d] + a[n/d]]; Array[a, 100] (* Jean-François Alcover, Feb 26 2016 *)