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A064097
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A quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.
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11
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0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 6, 6, 7, 6, 7, 5, 7, 6, 7, 6, 7, 7, 7, 6, 7, 7, 8, 7, 7, 8, 9, 6, 8, 7, 7, 7, 8, 7, 8, 7, 8, 8, 9, 7, 8, 8, 8, 6, 8, 8, 9, 7, 9, 8, 9, 7, 8, 8, 8, 8, 9, 8, 9, 7, 8, 8, 9, 8, 8, 9, 9, 8, 9, 8, 9, 9, 9, 10, 9, 7, 8, 9, 9, 8, 9, 8, 9, 8
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Note that this is the logarithm of a completely multiplicative function. - Michael Somos
Number of steps of iterations of {r - (largest divisor d < r)} needed to reach 1 starting at r = n. Example (a(19) = 6): 19 - 1 = 18; 18 - 9 = 9; 9 - 3 = 6; 6 - 3 = 3; 3 - 1 = 2; 2 - 1 = 1; iterations has 6 steps. a(n) = a(n - A032472(n)) + 1 for n >= 2. [From Jaroslav Krizek, Jan 28 2010]
See A105017, A000079, A175125. [From Jaroslav Krizek, Feb 15 2010]
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LINKS
| Hugo Pfoertner, Addition chains
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FORMULA
| Conjectures: for n>1, log(n) < a(n) < (5/2)*log(n); lim n ->infinity sum(k=1, n, a(k))/(n*log(n)-n) = C = 1.8(4)... - Benoit Cloitre, Oct 30 2002.
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MATHEMATICA
| quasiLog := (Length@NestWhileList[# - Divisors[#][[-2]] &, #, # > 1 &] - 1) &;
quasiLog /@ Range[1024]
(* Terentyev Oleg, July 17 2011 *)
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PROG
| (PARI) oo=200; an=vector(oo); a(n)=an[n]; for(n=2, oo, an[n]=if(isprime(n), 1+a(n-1), sumdiv(n, p, if(isprime(p), a(p)*valuation(n, p))))); for(n=1, 100, print1(a(n)", "))
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CROSSREFS
| Similar to A061373 which uses the same recurrence relation but a(1) = 1.
Cf. A003313, A076142, A076091, A061373, A005245.
For records see A105017.
Sequence in context: A003313 A117497 A117498 * A014701 A056239 A161511
Adjacent sequences: A064094 A064095 A064096 * A064098 A064099 A064100
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KEYWORD
| nonn
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AUTHOR
| Thomas Schulze (jazariel(AT)tiscalenet.it), Sep 16 2001
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EXTENSIONS
| More terms from Michael Somos, Sep 25 2001
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