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A179480

Let m>k>0 be odd numbers and denote by the symbol "m<->k" the value A000265(m-k). Then the sequence m<->k, m<->(m<->k), m<->(m<->(m<->k)), ... is periodic; a(n) is the smallest period in the case m=2*n-1, k=1.

1, 1, 2, 1, 3, 3, 2, 1, 5, 2, 6, 5, 5, 7, 2, 1, 6, 9, 6, 3, 3, 6, 12, 10, 4, 13, 10, 3, 15, 15, 2, 1, 17, 10, 18, 2, 10, 14, 20, 13, 21, 2, 14, 4, 6, 4, 18, 11, 9, 25, 26, 4, 27, 9, 18, 5, 22, 4, 12, 27, 10, 25, 2, 1, 33, 6, 18, 15, 35, 22, 30, 3, 22, 37, 6, 12, 10, 13, 26

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OFFSET

2,3

COMMENTS

A dual sequence to A179382.

Let b = (2*n-1) and k = A003558(n-1). If a(n) is odd, b divides (2^k + 1); but if a(n) is even, b divides (2^k - 1). Examples: a(14) = 5, odd; with b = 27 and A003558(13) = 9. Then 27 divides (2^9 + 1) or 513 = 27 * 19. a(18) = 6, even. b = 35, with k= A003558(17) = 12. Then 35 divides (2^12 - 1). - Gary W. Adamson, Aug 20 2012.

Iff a(n) = n/2 or (n-1)/2, then 2*n - 1 is a prime with one coach and is in A216371. Examples: a(19) = 9, so 37 is in A216371. a(12) = 6, so 23 is in A216371. - _Gary W. Adamson, Sep 08 2012.

LINKS

Table of n, a(n) for n=2..80.

EXAMPLE

If n=14, then m=27 and we have 27<->1=13, 27<->13=7, 27<->7=5, 27<->5=11, 27<->11=1. Thus a(14)=5.

MAPLE

Contribution from R. J. Mathar, Nov 04 2010: (Start)

A179480aux := proc(x, y) local xtrack, xitr, xpos ; xtrack := [y] ; while true do xitr := A000265(x-op(-1, xtrack)) ; if not member(xitr, xtrack, 'xpos') then xtrack := [op(xtrack), xitr] ; else return 1+nops(xtrack)-xpos ; end if; end do: end proc:

A179480 := proc(n) A179480aux(2*n-1, 1) ; end proc: seq(A179480(n), n=2..80) ; (End)

MATHEMATICA

oddres[n_] := n/2^IntegerExponent[n, 2];

b[x_, y_] := Module[{xtrack = {y}, xitr}, While[True, xitr = oddres[x - Last@ xtrack]; If[FreeQ[xtrack, xitr], AppendTo[xtrack, xitr], Return[ Length[xtrack]]]]];

a[n_] := b[2n-1, 1];

a /@ Range[2, 80] (* Jean-François Alcover, Apr 13 2020, after R. J. Mathar *)

CROSSREFS

Cf. A179382, A179383, A000265.

Cf. A003558.

Cf. A216371.

Sequence in context: A262209 A324338 A047679 * A245326 A241534 A337137

Adjacent sequences: A179477 A179478 A179479 * A179481 A179482 A179483

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Jul 16 2010

EXTENSIONS

Edited by N. J. A. Sloane, Jul 18 2010

More terms from R. J. Mathar, Nov 04 2010

STATUS

approved