A102421 -id:A102421 - cp's OEIS frontend

A102423 Start at x=2n+1, iterate the map x -> A337349(x); sequence gives smallest number in the resulting cycle, or -1 if the trajectory never cycles.

1, 1, 1, 1, 1, 1, 1, 17, 17, 17, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 17, 1, 1, 17, 17, 1, 1, 1, 1, 1, 1, 1, 1, 17, 17, 1, 1, 1, 1, 17, 17, 1, 1, 1, 1, 1, 17, 17, 1, 1, 1, 1, 1, 17, 17, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 17, 17, 17, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, based on email from Dan Asimov (dasimov(AT)earthlink.net), Sep 15 2006

nonn

See A102421 and A337349 for further comments.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A102421, A122563 (iterations to enter cycle), A337349.

nextx[x_Integer] := Block[{a = x}, a = 3a + 1; While[EvenQ@a, a /= 2]; a = 3a - 1; While[EvenQ@a, a /= 2]; a]; f[n_] := NestWhile[nextx, n, FreeQ[{1, 17, 19, 43, 97, 109, 61}, # ] &]; Table[ If[ f[2n + 1] == 1, 1, 17], {n, 0, 94}] (* Robert G. Wilson v, Sep 20 2006; added 61 to comparison set used for detecting cycles, William P. Orrick, Aug 24 2020; should be "!" prefixing MemberQ, changed to FreeQ, Ray Chandler, Aug 28 2020 *)

More terms from Robert G. Wilson v, Sep 20 2006

Replaced A102421 by A337349 in NAME. - R. J. Mathar, Aug 24 2020

A337349 To get a(n), take 3*n+1 and divide out any power of 2; then multiply by 3, subtract 1 and divide out any power of 2.

Original entry on oeis.org

1, 1, 5, 7, 19, 1, 7, 1, 37, 5, 23, 25, 55, 7, 1, 17, 73, 19, 41, 43, 91, 1, 25, 13, 109, 7, 59, 61, 127, 1, 17, 35, 145, 37, 77, 79, 163, 5, 43, 11, 181, 23, 95, 97, 199, 25, 13, 53, 217, 55, 113, 115, 235, 7, 61, 31, 253, 1, 131, 133, 271, 17, 35, 71, 289, 73, 149

Offset: 0

N. J. A. Sloane, based on email from Dan Asimov (dasimov(AT)earthlink.net), Sep 15 2006

When a(x) is iterated, what are the limit cycles? Are there any besides (1) and (17 -> 19 -> 43 -> 97 -> 109 -> 61)?

Up to 1000000000 every number eventually reaches one of those two cycles. In this range, the longest trajectory starts with n=458788881 and takes 193 steps to reach 1. - Christian Boyer (cboyer(AT)club-internet.fr), Sep 16 2006

Hugo Pfoertner, Table of n, a(n) for n = 0..10000

Cf. A102421 (bisection), A102423.

A337349 := proc(n)
    local a;
    a := 3*n+1;
    while modp(a,2) = 0 do
        a := a/2 ;
    end do:
    a := 3*a-1 ;
    while modp(a,2) = 0 do
        a := a/2 ;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 24 2020

a[n_] := Module[{k = 3n+1}, k = k/2^IntegerExponent[k, 2]; k = 3k-1; k = k/2^IntegerExponent[k, 2]; k];
a /@ Range[0, 100] (* Jean-François Alcover, Aug 27 2020 *)

a(n) = A075677(A067745(n+1)).

a(2*n+1) = A102421(n).

A122563 Start at 2n+1, iterate the map x -> A337349(x); sequence gives the number of iterations to resulting cycle or -1 if the trajectory never cycles.

Original entry on oeis.org

0, 2, 1, 1, 2, 3, 2, 1, 0, 0, 1, 3, 2, 1, 1, 6, 3, 5, 2, 4, 4, 0, 3, 3, 8, 8, 2, 7, 1, 4, 0, 3, 6, 3, 1, 2, 5, 10, 1, 4, 10, 7, 1, 9, 3, 9, 3, 8, 0, 8, 2, 2, 5, 7, 0, 7, 7, 7, 1, 4, 1, 2, 6, 6, 6, 9, 3, 1, 2, 5, 5, 5, 5, 8, 2, 2, 1, 10, 4, 16, 4, 4, 4, 4, 9, 6, 1, 9, 3, 15, 3, 3, 3, 6, 3, 3, 2, 8, 8, 2, 8, 14

Offset: 0

Robert G. Wilson v, based on email from Dan Asimov (dasimov(AT)earthlink.net), Sep 20 2006

Iteration: multiply by 3 and add 1 and divide out any power of 2; then multiply by 3 and subtract 1 and divide out any power of 2.

When a(x) is iterated, what are the limit cycles? Are there any besides {1} and {17 -> 19 -> 43 -> 97 -> 109 -> 61}?

			The iteration for n=13 is 27->61->17->19->43->97->109->61->... and a(13)=1 step was needed to enter the cycle (at 61).
The iteration for n=30 is 61-> 17->19->43->97->109->61->> and the cycle was already entered at the start, so a(30)=0.

Ray Chandler, Table of n, a(n) for n = 0..10000

Cf. A102421, A102423, A337349.

A122563 := proc(n)
    local cyc,itr,x ;
    cyc := [] ;
    x := 2*n+1 ;
    while true do
        cyc := [op(cyc),x] ;
        x := A337349(x) ;
        if x in cyc then
            break ;
        end if;
    end do:
    member(x,cyc,'itr') ;
    itr -1 ;
end proc:
seq(A122563(n),n=0..101) ; # R. J. Mathar, Aug 26 2020

nextx[x_Integer] := Block[{a = x}, a = 3 a + 1; While[EvenQ@a, a /= 2]; a = 3 a - 1; While[EvenQ@a, a /= 2]; a]; f[n_] := Length@NestWhileList[nextx, n, FreeQ[{1, 17, 19, 43, 97, 109, 61}, #] &] - 1; Table[f[2 n + 1], {n, 0, 101}] (* original program from author corrected as suggested by William P. Orrick, Ray Chandler, Aug 28 2020 *)

a(13), a(30),... corrected. - R. J. Mathar, Aug 26 2020

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A102423 Start at x=2n+1, iterate the map x -> A337349(x); sequence gives smallest number in the resulting cycle, or -1 if the trajectory never cycles.

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A337349 To get a(n), take 3*n+1 and divide out any power of 2; then multiply by 3, subtract 1 and divide out any power of 2.

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A122563 Start at 2n+1, iterate the map x -> A337349(x); sequence gives the number of iterations to resulting cycle or -1 if the trajectory never cycles.

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