A175764 Number of iterations of the mapping k->f(k) to reach one of 2, 5, or 29, starting with k=n, and with f(k)=(k^2+4)/d, where d is the next-to-largest divisor of k^2+4, or -1 if the sequence never reaches one of the required values.

1, 0, 9, 1, 0, 1, 2, 1, 1, 1, 1, 1, 8, 1, 2, 1, 4, 1, 1, 1, 1, 1, 9, 1, 5, 1, 3, 1, 0, 1, 1, 1, 3, 1, 2, 1, 6, 1, 1, 1, 1, 1, 5, 1, 2, 1, 10, 1, 1, 1, 1, 1, 1, 1, 9, 1, 10, 1, 1, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 1, 1, 10, 1, 9, 1, 5, 1, 1, 1, 1, 1, 2, 1, 2, 1, 6, 1, 1, 1, 1, 1, 5, 1, 2, 1, 3, 1, 1, 1, 1, 1, 11

Offset

1,3

Comments

It appears that the sequence always reaches 2, 5, or 29 for any initial value n. Is this easy to prove?

It appears that a(n) is 1 whenever n>29 and n mod 10 is one of {0,1,2,4,6,8,9}. This has been verified to n=5000. Also, it appears that a(n) is 9 whenever n mod 130 is one of {3,23,55,75,107,127}. This has also been verified to n=5000. Are these conjectures easy to prove?

From Antti Karttunen, May 19 2021: (Start)

Note that the first four terms of iteration 47017 -> 2210598293 -> 4886744813014513853 -> 23880274867524255960728999629928905613 are all primes (see also A231120), but then (4+(23880274867524255960728999629928905613^2)) is composite, and its smallest prime divisor is 1946761. Actually, a(23880274867524255960728999629928905613) = 2, thus a(47017) = 5.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..47016

Example

For n=3, we have 3 -> (3^2+4)/d = 13/1 -> (13^2+4)/d = 173/1 -> (173^2+4)/d = 29933/809 = 37, since the divisors of 29933 are {1,37,809,29933}. Continuing, we get the orbit {3,13,173,37,1373,1217,97,9413,89,5,29,5,29,...}, showing that 5 is reached after 9 steps, after which the orbit is periodic {...,5,29,5,29,...}. Thus a(3)=9.

Prog

(PARI)
A175764(n) = if(2==n||5==n||29==n, 0, 1+A175764(f(n)));
f(k) = { my(u=(4+(k^2)), ds=divisors(u)); (u/ds[#ds-1]); };
\\ Alternatively, "f" could be defined as:
f(k) = { my(u=(4+(k^2))); (u/A032742(u)); };
A032742(n) = if(1==n||isprime(n), 1, forprime(p=2, n, if(!(n%p), return(n/p)))); \\ And not requiring full factorization when this is used. - Antti Karttunen, May 19 2021

Crossrefs

Cf. A032742, A076423, A087717, A231120.

Sequence in context: A155783 A257097 A256667 * A269948 A121935 A070060

Adjacent sequences: A175761 A175762 A175763 * A175765 A175766 A175767

Keyword

nonn

Author

John W. Layman, Aug 30 2010

Status

approved