A063041 Image of n under Collatz-2 map, a generalization of the classical '3x+1' - function: instead of dividing an even number by 2 a nonprime will be divided by its smallest prime factor and a prime will be multiplied not by 3 but by its prime-predecessor, before one is added.

3, 7, 2, 16, 3, 36, 4, 3, 5, 78, 6, 144, 7, 5, 8, 222, 9, 324, 10, 7, 11, 438, 12, 5, 13, 9, 14, 668, 15, 900, 16, 11, 17, 7, 18, 1148, 19, 13, 20, 1518, 21, 1764, 22, 15, 23, 2022, 24, 7, 25, 17, 26, 2492, 27, 11, 28, 19, 29, 3128, 30, 3600, 31, 21, 32, 13, 33, 4088, 34, 23

Offset

2,1

Links

Antti Karttunen, Table of n, a(n) for n = 2..10001

Formula

a(n) = if n prime then (n * pp(n) + 1) else (n / lpf(n)) for n > 1 where pp(n) = if n > 2 then Max{p prime | p < n} else 1; [prime-predecessor] and lpf(n) = if n > 2 then Min{p prime | p < n and p divides n} else 1; [where lpf = A020639].

If A010051(n) = 1 [when n is a prime], a(n) = 1 + (A064989(n)*n), otherwise a(n) = A032742(n). - Antti Karttunen, Jan 23 2017

Example

a(17) = 17 * 13 = 222 as 17 is prime and 13 is the largest prime < 17; a(4537) = 349 as 4537 = 13 * 349 hence lpf(4537) = 13; other examples in A063042, A063043, A063044.
For n=2, its prime-predecessor is taken as 1 (because 2 is the first prime), thus a(2) = (1*2)+1 = 3.

Mathematica

Join[{3}, Table[If[PrimeQ[n], n*Prime[PrimePi[n]-1]+1, n/Min[First/@FactorInteger[n]]], {n, 3, 69}]] (* Jayanta Basu, May 27 2013 *)

Prog

(Scheme) (define (A063041 n) (if (= 1 (A010051 n)) (+ 1 (* (A064989 n) n)) (A032742 n))) ;; Antti Karttunen, Jan 23 2017
(Python)
from sympy import isprime, prevprime, primefactors
def f(n): return 1 if n == 2 else prevprime(n)
def a(n): return n*f(n)+1 if isprime(n) else n//min(primefactors(n))
print([a(n) for n in range(2, 70)]) # Michael S. Branicky, Apr 17 2023

Crossrefs

Cf. A010051, A020639, A032742, A063045, A063046, A064989.

Cf. A063042, A063043, A063044, A280707 (trajectories starting from 3, 17, 29 and 47).

Sequence in context: A318467 A324713 A245611 * A257729 A328502 A308480

Adjacent sequences: A063038 A063039 A063040 * A063042 A063043 A063044

Keyword

nonn,easy

Author

Reinhard Zumkeller, Jul 07 2001

Extensions

More terms from Matthew Conroy, Jul 15 2001

Description clarified by Antti Karttunen, Jan 23 2017

Status

approved