

A075677


Reduced Collatz function R applied to the odd integers: a(n) = R(2n1), where R(k) = (3k+1)/2^r, with r as large as possible.


27



1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53, 7, 59, 31, 65, 17, 71, 37, 77, 5, 83, 43, 89, 23, 95, 49, 101, 13, 107, 55, 113, 29, 119, 61, 125, 1, 131, 67, 137, 35, 143, 73, 149, 19, 155, 79, 161, 41, 167, 85, 173, 11, 179, 91, 185, 47, 191, 97, 197
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OFFSET

1,2


COMMENTS

The evenindexed terms a(2i+2) = 6i+5 = A016969(i), i >= 0 [Comment corrected by Bob Selcoe, Apr 06 2015]. The oddindexed terms terms are the same as A067745. Note that this sequence is A016789 with all factors of 2 removed from each term. Also note that a(4i1) = a(i). No multiple of 3 is in this sequence. See A075680 for the number of iterations of R required to yield 1.
From Bob Selcoe, Apr 06 2015: (Start)
All numbers in this sequence appear infinitely often.
From Eq. 1 and Eq. 2 in Formulas: Eq. 1 is used with 1/3 of the numbers in this sequence, Eq. 2 is used with 2/3 of the numbers.
(End)
Empirical: For arbitrary m, Sum_{n=2..A007583(m)} (A075677(n)  A075677(n1)) = 0.  Fred Daniel Kline, Nov 23 2015


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, E16.
Victor Klee and Stan Wagon, Old and new unsolved problems in plane geometry and number theory, The Mathematical Association of America, 1991, p. 225, C(2n+1) = a(n+1), n >= 0.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see p. 57, also (909), p. 306.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Chamberland, Una actualizacio del problema 3x + 1, Butl. Soc. Catalana Mat. (18), 1945, 2003.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 323.
Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for sequences related to 3x+1 (or Collatz) problem


FORMULA

a(n) = A000265(6*n2) = A000265(3*n1).  Reinhard Zumkeller, Jan 08 2014
From Bob Selcoe, Apr 05 2015: (Start)
For all n>=1 and for every k, there exists j>=0 dependent upon n and k such that either:
Eq. 1: a(n) = (3n1)/2^(2j+1) when k = ((4^(j+1)1)/3) mod 2^(2j+3). Alternatively: a(n) = A016789(n1)/A081294(j+1) when k = A002450(j+1) mod A081294(j+2). Example: n=51; k=101 == 5 mod 32, j=1. a(51) = 152/8 = 19.
or
Eq. 2: a(n) = (3n1)/4^j when k = (5*2^(2j+1)  1)/3) mod 4^(j+1). Alternatively: a(n) = A016789(n1)/A000302(j) when k = A072197(j) mod A000302(j+1). Example: n=91; k=181 == 53 mod 64, j=2. a(91) = 272/16 = 17.
(End) [Definition corrected by William S. Hilton, Jul 29 2017]
a(n) = a(n + g*2^r)  6*g, n > g*2^r. Examples: n=59; a(59)=11, r=5. g=1: 11 = a(27) = 5  (1)*6; g=1: 11 = a(91) = 17  1*6; g=2: 11 = a(123) = 23  2*6; g=3: 11 = a(155) = 29  3*6; etc.  Bob Selcoe, Apr 06 2015
a(n) = a((1 + (3*n  1)*4^(k1))/3), k>=1 (cf. A191669).  L. Edson Jeffery, Oct 05 2015
a(n) = a(4n1).  Bob Selcoe, Aug 03 2017


EXAMPLE

a(11) = 1 because 21 is the 11th odd number and R(21) = 64/64 = 1.


MAPLE

f:=proc(n) local t1;
if n=1 then RETURN(1) else
t1:=3*n+1;
while t1 mod 2 = 0 do t1:=t1/2; od;
RETURN(t1); fi;
end;
# N. J. A. Sloane, Jan 21 2011


MATHEMATICA

nextOddK[n_] := Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]
v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; Table[f[2*n  1], {n, 66}] (* L. Edson Jeffery, May 06 2015 *)


PROG

(PARI) a(n)=n+=2*n1; n>>valuation(n, 2) \\ Charles R Greathouse IV, Jul 05 2013
(Haskell)
a075677 = a000265 . subtract 2 . (* 6)  Reinhard Zumkeller, Jan 08 2014
(Python)
from sympy import divisors
def a(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
print [a(6*n  2) for n in range(1, 101)] # Indranil Ghosh, Apr 15 2017, after formula by Reinhard Zumkeller


CROSSREFS

Cf. A000265, A000302, A002450, A016789, A016969, A065677, A072197, A075680, A081294.
Sequence in context: A185953 A323359 A324036 * A051853 A159074 A147414
Adjacent sequences: A075674 A075675 A075676 * A075678 A075679 A075680


KEYWORD

easy,nonn,changed


AUTHOR

T. D. Noe, Sep 25 2002


STATUS

approved



