A006877 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.

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

Offset: 1

Author

N. J. A. Sloane and Robert Munafo

Keywords

• nonn
• nice

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

• Hugo Pfoertner, Table of n, a(n) for n = 1..148 (from Eric Rosendaal's 3x+1 Delay Records, terms 1..130 from T. D. Noe)
• T. Ahmed and H. Snevily, Are there an infinite number of Collatz integers?, 2013.
• 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).
• 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)
• Robert Munafo, Integer Sequences Related to 3x+1 Collatz Iteration
• Eric Roosendaal, 3x+1 Delay Records
• Olivier Rozier and Claude Terracol, Paradoxical behavior in Collatz sequences, arXiv:2502.00948 [math.GM], 2025. See p. 15.
• Robert G. Wilson v, Letter to N. J. A. Sloane with attachments, Jan. 1989
• Robert G. Wilson v, Tables of A6877, A6884, A6885, Jan. 1989
• Index entries for sequences from "Goedel, Escher, Bach"
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006884, A006885, A006877, A006878, A033492, A380138.

Programs

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 *)

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