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A160266
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Let f and its k-fold iteration f^k be defined as in A159885. a(n) is the least k for which A006694( (f^k(2n+1)-1)/2 ) < A006694(n).
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7
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2, 1, 1, 2, 4, 2, 1, 1, 6, 1, 2, 1, 1, 5, 1, 1, 1, 6, 1, 4, 3, 1, 2, 1, 1, 2, 1, 1, 10, 5, 1, 1, 8, 1, 1, 1, 1, 1, 2, 1, 40, 1, 1, 1, 1, 1, 6, 3, 1, 7, 17, 1, 36, 1, 1, 2, 1, 1, 1, 20, 1, 1, 1, 1, 8, 1, 1, 18, 13, 1, 5, 1, 2, 6, 1, 1, 1, 1, 1, 1, 6, 1, 9, 11, 2, 9, 1, 2, 9, 4, 6, 1, 1, 1, 9, 7, 1, 7, 29, 2, 2, 1
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OFFSET
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1,1
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COMMENTS
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Conjecture. For every n>=1, there exists a finite value of a(n). It is easy to see that this conjecture is equivalent to the well known Collatz 3n+1 conjecture.
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LINKS
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MAPLE
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A006519 := proc(n) local i ; for i in ifactors(n)[2] do if op(1, i) = 2 then return op(1, i)^op(2, i) ; fi ; od: return 1 ; end proc:
f := proc(twon1) local threen2 ; threen2 := 3*twon1/2+1/2 ; threen2/A006519(threen2) ; end proc:
A160266 := proc(n) local ref, k, fk ; ref := A006694(n) ; k := 1 ; fk := f(2*n+1) ; while true do if A006694( (fk-1)/2 ) < ref then return k; end if; fk := f(fk) ; k := k+1 ; end do ; end proc:
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PROG
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(PARI)
f(n) = ((3*((n-1)/2))+2)/A006519((3*((n-1)/2))+2);
A006694(n) = (sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1); \\ From A006694
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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