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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. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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]

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-Fran├žois Alcover, Feb 26 2016 *)

PROG

(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

CROSSREFS

Cf. A005245.

Cf. A020639.

Sequence in context: A091333 A293771 A005245 * A327705 A104135 A276656

Adjacent sequences:  A061370 A061371 A061372 * A061374 A061375 A061376

KEYWORD

easy,nice,nonn

AUTHOR

Juan Arias-de-Reyna, Jun 08 2001

STATUS

approved

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Last modified December 11 23:44 EST 2019. Contains 329945 sequences. (Running on oeis4.)