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A006877 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.
(Formerly M0748)
27
1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Both the 3x+1 steps and the halving steps are counted.
This sequence without a(2) = 2 specifies where records occur in A208981. - Omar E. Pol, Apr 14 2022
REFERENCES
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..130 (from Eric Roosendaal's data)
Gaston H. Gonnet, Computations on the 3n+1 conjecture, Maple Technical Newsletter 6 (1991): 18-22.
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Eric Roosendaal, 3x+1 Delay Records
David Barina, Convergence verification of the Collatz problem, The Journal of Supercomputing 77(3) (2021), 2681-2688.
David Barina, Computational Verification of the Collatz Problem, preprint on Research Square (2024).
MAPLE
A006877 := proc(n) local a, L; L := 0; a := n; while a <> 1 do if a mod 2 = 0 then a := a/2; else a := 3*a+1; fi; L := L+1; od: RETURN(L); end;
MATHEMATICA
numberOfSteps[x0_] := Block[{x = x0, nos = 0}, While [x != 1 , If[Mod[x, 2] == 0 , x = x/2, x = 3*x + 1]; nos++]; nos]; a[1] = 1; a[n_] := a[n] = Block[{x = a[n-1] + 1}, record = numberOfSteps[x - 1]; While[ numberOfSteps[x] <= record, x++]; x]; A006877 = Table[ Print[a[n]]; a[n], {n, 1, 44}](* Jean-François Alcover, Feb 14 2012 *)
DeleteDuplicates[Table[{n, Length[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]]}, {n, 838000}], GreaterEqual[#1[[2]], #2[[2]]]&][[All, 1]] (* Harvey P. Dale, May 13 2022 *)
PROG
(PARI) A006577(n)=my(s); while(n>1, n=if(n%2, 3*n+1, n/2); s++); s
step(n, r)=my(t); forstep(k=bitor(n, 1), 2*n, 2, t=A006577(k); if(t>r, return([k, t]))); [2*n, r+1]
r=0; print1(n=1); for(i=1, 100, [n, r]=step(n, r); print1(", "n)) \\ Charles R Greathouse IV, Apr 01 2013
(Python)
c1 = lambda x: (3*x+1 if (x%2) else x>>1)
r = -1
for n in range(1, 10**5):
a=0 ; n1=n
while n>1: n=c1(n); a+=1;
if a > r: print(n1, end = ', '); r=a
print('...') # Ya-Ping Lu and Robert Munafo, Mar 22 2024
CROSSREFS
Sequence in context: A018700 A018295 A033495 * A328832 A263881 A208892
KEYWORD
nonn,nice
AUTHOR
STATUS
approved

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Last modified May 25 07:15 EDT 2024. Contains 372782 sequences. (Running on oeis4.)