A334097 -id:A334097 - cp's OEIS frontend

A334098 a(n) = A334097(n) - A331410(n), where former is the exponent of the eventual power of 2 reached, and the latter is the number of iterations needed to get there, when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.

0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 3, 4, 2, 3, 4, 5, 3, 3, 3, 4, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 4, 3, 3, 4, 3, 4, 3, 5, 3, 3, 3, 4, 4, 5, 4, 6, 3, 4, 3, 4, 4, 4, 4, 5, 2, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 5, 3, 4, 3, 5, 3, 4, 4, 5, 5, 5, 3, 6, 4, 5, 4, 4, 3, 4, 4, 5, 4

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Question: Are there any negative terms?

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A331410, A334097.

Array[Log2@ Last[#] - (Length[#] - 1) &@ NestWhileList[# + #/FactorInteger[#][[-1, 1]] &, #, ! IntegerQ@ Log2@ # &] &, 105] (* Michael De Vlieger, Apr 30 2020 *)

A334098(n) = { my(k=0); while(bitand(n,n-1), k++; my(f=factor(n)[, 1]); n += (n/f[2-(n%2)])); (valuation(n,2)-k); };

a(n) = A334097(n) - A331410(n).

Totally additive sequence: a(m*n) = a(m)+a(n), for all m, n.

A334862 a(n) = A334097(n) - A064415(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 2, 1, 1, 3, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 3, 2, 1, 2, 1, 3, 1, 1, 2, 3, 2, 2, 2, 1, 1, 3

Offset: 1

Antti Karttunen, May 14 2020

nonn

Completely additive because A064415 and A334097 are.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079 (positions of zeros), A000244, A064415, A334097, A334861.

A064415(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],f[k,2],f[k,2]*A064415(f[k,1]-1))); };
A334097(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],f[k,2],f[k,2]*A334097(f[k,1]+1))); };
A334862(n) = (A334097(n)-A064415(n));
\\ Or alternatively as:
A334862(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(A334097(f[k,1]+1)-A064415(f[k,1]-1)))); };

a(2) = 0, a(p) = A334097(p+1)-A064415(p-1) for odd primes p, a(m*n) = a(m)+a(n), if m,n > 1.
a(n) = A334097(n) - A064415(n).
a(3^k) = k for all k>= 0.

A331410 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k + k/p, where p is the largest prime factor of k.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 2, 2, 2, 1, 2, 1, 3, 0, 3, 2, 3, 2, 2, 2, 2, 1, 4, 2, 3, 1, 4, 3, 1, 0, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 2, 1, 2, 4, 4, 2, 4, 3, 4, 1, 4, 4, 4, 3, 2, 1, 3, 0, 4, 3, 4, 3, 3, 3, 3, 2, 5, 4, 5, 3, 3, 3, 3, 2, 4, 3, 3, 2, 5, 3, 5, 2, 5, 4, 3, 2, 2, 2, 5, 1, 3, 2, 4, 4, 5, 4, 3, 2, 4

Offset: 1

Ali Sada, Jan 16 2020

nonn

Let f(n) = A000265(n) be the odd part of n. Let p be the largest prime factor of k, and say k = p * m. Suppose that k is not a power of 2, i.e., p > 2, then f(k) = p * f(m). The iteration is k -> k + k/p = p*m + m = (p+1) * m. So, p * f(m) -> f(p+1) * f(m). Since for p > 2, f(p+1) < p, the odd part in each iteration decreases, until it becomes 1, i.e., until we reach a power of 2. - Amiram Eldar, Feb 19 2020
Any odd prime factor of k can be used at any step of the iteration, and the result will be same. Thus, like A329697, this is also fully additive sequence. - Antti Karttunen, Apr 29 2020
If and only if a(n) is equal to A005087(n), then sigma(2n) - sigma(n) is a power of 2. (See A336923, A046528). - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is [15,18,24,32], taking 3 iterations to reach 32. So, a(15) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree of a(n) labeling the index n and applying a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value of a(n) for n = 1..16383.

Cf. A000265, A005087, A006530 (greatest prime factor), A052126, A078701, A087436, A329662 (positions of records and the first occurrences of each n), A334097, A334098, A334108, A334861, A336467, A336921, A336922, A336923 (A046528).
Cf. array A335430, and its rows A335431, A335882, and also A335874.
Cf. also A329697 (analogous sequence when using the map k -> k - k/p), A335878.
Cf. also A330437, A335884, A335885, A336362, A336363 for other similar iterations.

f:=func; g:=func; a:=[]; for n in [1..1000] do k:=n; s:=0; while not g(k) do  s:=s+1; k:=f(k); end while; Append(~a,s); end for; a; // Marius A. Burtea, Jan 19 2020

a[n_] := -1 + Length @ NestWhileList[# + #/FactorInteger[#][[-1, 1]] &, n, # / 2^IntegerExponent[#, 2] != 1 &]; Array[a, 100] (* Amiram Eldar, Jan 16 2020 *)

A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 29 2020

A331410(n) = { my(k=0); while(bitand(n,n-1), k++; my(f=factor(n)[, 1]); n += (n/f[2-(n%2)])); (k); }; \\ Antti Karttunen, Apr 29 2020

A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(1+f[k,1])))); }; \\ Antti Karttunen, Apr 30 2020

From Antti Karttunen, Apr 29 2020: (Start)
This is a completely additive sequence: a(2) = 0, a(p) = 1+a(p+1) for odd primes p, a(m*n) = a(m)+a(n), if m,n > 1.
a(2n) = a(A000265(n)) = a(n).
If A209229(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(n+A052126(n)), or equally, 1 + a(n+(n/A078701(n))).
a(n) = A334097(n) - A334098(n).
a(A122111(n)) = A334108(n).
(End)
a(n) = A334861(n) - A329697(n). - Antti Karttunen, May 14 2020
a(n) = a(A336467(n)) + A087436(n) = A336921(n) + A087436(n). - Antti Karttunen, Mar 16 2021

Data section extended up to a(105) by Antti Karttunen, Apr 29 2020

A064415 a(1) = 0, a(n) = iter(n) if n is even, a(n) = iter(n)-1 if n is odd, where iter(n) = A003434(n) = smallest number of iterations of Euler totient function phi needed to reach 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 4, 6, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 6, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6, 5

Offset: 1

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

nonn

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]

Cf. A000010, A003434, A007814, A052126, A054725, A064097, A064416, A064674, A171462, A291783, A329697.
The 2-adic valuation of A309243.
Partial sums of A334195. Cf. A053044 for partial sums of this sequence.
Cf. also A334097 (analogous sequence when using the map k -> k + k/p).

a064415 1 = 0
a064415 n = a003434 n - n `mod` 2 -- Reinhard Zumkeller, Sep 18 2011

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

A064415(n) = if(!bitand(n,n-1),valuation(n,2),A064415(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, May 13 2020

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

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)
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)). [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(n) = A064097(n) - A329697(n).
a(2^k) = a(3^k) = k.
(End)

More terms from David Wasserman, Jul 22 2002

Definition corrected by Reinhard Zumkeller, Sep 18 2011

A335876 a(1) = 2, and for n > 1, a(n) = n + (n/p), where p is largest prime dividing n, A006530(n).

Original entry on oeis.org

2, 3, 4, 6, 6, 8, 8, 12, 12, 12, 12, 16, 14, 16, 18, 24, 18, 24, 20, 24, 24, 24, 24, 32, 30, 28, 36, 32, 30, 36, 32, 48, 36, 36, 40, 48, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 64, 56, 60, 54, 56, 54, 72, 60, 64, 60, 60, 60, 72, 62, 64, 72, 96, 70, 72, 68, 72, 72, 80, 72, 96, 74, 76, 90, 80, 84, 84, 80, 96, 108, 84, 84

Offset: 1

Antti Karttunen, Jun 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006530, A052126, A171462, A331410, A334097, A335431 (positions of two's powers > 2).

Array[# (1 + 1/FactorInteger[#][[-1, 1]]) &, 83] (* Michael De Vlieger, Jul 08 2020 *)

A335876(n) = if(1==n,2,n + (n/vecmax(factor(n)[, 1])));

a(n) = n + A052126(n).

cp's OEIS Frontend

A334098 a(n) = A334097(n) - A331410(n), where former is the exponent of the eventual power of 2 reached, and the latter is the number of iterations needed to get there, when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.

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A334862 a(n) = A334097(n) - A064415(n).

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A331410 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k + k/p, where p is the largest prime factor of k.

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A064415 a(1) = 0, a(n) = iter(n) if n is even, a(n) = iter(n)-1 if n is odd, where iter(n) = A003434(n) = smallest number of iterations of Euler totient function phi needed to reach 1.

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A335876 a(1) = 2, and for n > 1, a(n) = n + (n/p), where p is largest prime dividing n, A006530(n).

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