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.

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

Author

Ali Sada, Jan 16 2020

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

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.

Programs

Magma

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

Mathematica

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

PARI

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

PARI

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

PARI

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

Formula

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

Extensions

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