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A063041
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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.
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7
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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
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OFFSET
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2,1
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LINKS
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FORMULA
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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].
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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