

A061373


"Natural" logarithm, defined inductively by a(1)=1, a(p)=1+a(p1) 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
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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), 230250. (In Spanish.)
J. Arias de Reyna, Complejidad de los numeros naturales, Gaceta de la Real Sociedad Matematica Espanola, 3, (2000), 230250. (In Spanish.) [Cached copy, with permission]


MATHEMATICA

a[1]=1; a[p_?PrimeQ] := 1+a[p1]; a[n_] := a[n] = With[{d=Divisors[n][[2]] }, a[d] + a[n/d]]; Array[a, 100] (* JeanFrançois Alcover, Feb 26 2016 *)


PROG

(Haskell)
import Data.List (genericIndex)
a061373 1 = 1
a061373 n = genericIndex a061373_list (n1)
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 AriasdeReyna, Jun 08 2001


STATUS

approved



