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
Examples
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.
Links
- Antti Karttunen, Table of n, a(n) for n = 2..10001
Crossrefs
Programs
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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 *) -
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
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Scheme
(define (A063041 n) (if (= 1 (A010051 n)) (+ 1 (* (A064989 n) n)) (A032742 n))) ;; Antti Karttunen, Jan 23 2017
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
Extensions
More terms from Matthew Conroy, Jul 15 2001
Description clarified by Antti Karttunen, Jan 23 2017