OFFSET
1,4
COMMENTS
a(n) is the exponent of the eventual power of 2 reached when starting from k=n and then iterating the nondeterministic map k -> k-(k/p), where p can be any odd prime factor of k, for example, the largest. Note that each original odd prime factor p of n brings its own share of 2's to the final result after it has been completely processed (with all intermediate odd primes also eliminated, leaving only 2's). As no 2's are removed, also all 2's already present in the original n are included in the eventual power of 2 that is reached, implying that a(n) >= A007814(n). - Antti Karttunen, May 13 2020
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
FORMULA
For all integers m >0 and n>0 a(m*n)=a(m)+a(n). The function a(n) is completely additive. The smallest integer q which satisfy the equation a(q)=n is 2^q, the greatest is 3^q. For all integers n>0, the counter image off n, a^-1(n) is finite.
a(1) = 0 and a(n) = A054725(n) for n>=2. - Joerg Arndt, Apr 08 2014, A-number corrected by Antti Karttunen, May 13 2020
From Antti Karttunen, May 13 2020: (Start)
a(1) = 0, a(2) = 1 and for n > 2, a(n) = sum(p | n, a(p-1)), where sum is over all primes p that divide n, with multiplicity. (Cf. A054725).
a(1) = 0, a(2) = 1 and a(p) = 1 + a((p-1)/2) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [From above formula, 1+ compensates for the "lost" 2]
a(n) = A007814(A309243(n)). [From Rémy Sigrist's conjecture in the latter sequence. This reduces to a(n) = sum(p|n, a(p-1)) formula above, thus holds also]
If A209229(n) = 1 [when n is a power of 2], a(n) = A007814(n), otherwise a(n) = a(n-A052126(n)) = a(A171462(n)). [From the definition in the comments]
a(2^k) = a(3^k) = k.
(End)
PROG
(Haskell)
a064415 1 = 0
a064415 n = a003434 n - n `mod` 2 -- Reinhard Zumkeller, Sep 18 2011
(PARI)
A003434(n)=for(k=0, n, n>1 || return(k); n=eulerphi(n));
a(n) = if( n<=2, n-1, A003434(n) - (n%2) );
vector(200, n, a(n)) \\ Joerg Arndt, Apr 08 2014
(PARI) A064415(n) = if(!bitand(n, n-1), valuation(n, 2), A064415(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, May 13 2020
(PARI) A064415(n) = if(1==n, 0, if(isprime(n), 1+A064415(n>>1), my(f=factor(n)); (apply(A064415, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, May 13 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian WEINSBERG (cweinsbe(AT)fr.packardbell.org), Sep 30 2001
EXTENSIONS
More terms from David Wasserman, Jul 22 2002
Definition corrected by Reinhard Zumkeller, Sep 18 2011
STATUS
approved