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

1,5

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

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.

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

Example

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

Mathematica

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

Prog

(Magma) f:=func<n|n+n div p where p is Max(PrimeDivisors(n))>; g:=func<n| n eq 1 or Max(PrimeDivisors(n)) eq 2>; 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
(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

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.

Sequence in context: A123331 A235141 A347386 * A336928 A366388 A114638

Adjacent sequences: A331407 A331408 A331409 * A331411 A331412 A331413

Keyword

nonn

Author

Ali Sada, Jan 16 2020

Extensions

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

Status

approved