1,4

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

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]

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 byAntti Karttunen,May 13 2020

FromAntti Karttunen,May 13 2020: (Start)

For n > 1, a(n) =A003434(n) -A000035(n).

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)). [FromRémy Sigrist's conjecture in the latter sequence. This reduces to a(n) = sum(p|n, a(p-1)) formula above, thus holds also]

IfA209229(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(n) =A064097(n) -A329697(n).

a(2^k) = a(3^k) = k.

(End)

(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

Cf.A000010,A003434,A007814,A052126,A054725,A064097,A064416,A064674,A171462,A291783,A329697.

The 2-adic valuation ofA309243.

Partial sums ofA334195.Cf.A053044for partial sums of this sequence.

Cf. alsoA334097(analogous sequence when using the map k -> k + k/p).

Sequence in context:A241216A125173A054725*A086833A010764A029383

Adjacent sequences:A064412A064413A064414*A064416A064417A064418

Christian WEINSBERG (cweinsbe(AT)fr.packardbell.org), Sep 30 2001

More terms fromDavid Wasserman,Jul 22 2002

Definition corrected byReinhard Zumkeller,Sep 18 2011

approved