A221469 -id:A221469 - cp's OEIS frontend

A221470 Least m such that the Collatz (3x+1) iteration of m has exactly n increasing peak values.

1, 5, 3, 7, 15, 287, 191, 127, 223, 159, 143, 95, 63, 47, 31, 27, 703, 6471, 6383, 4255, 6887, 4591, 50427, 47867, 31911, 77671, 161439, 113383, 239231, 159487, 1027431, 974079, 730559, 487039, 432923, 288615, 270271, 3041391, 9158655, 6416623, 16786431, 12589823

Offset: 0

T. D. Noe, Jan 17 2013

nonn

Sequence A221469 lists the number of increasing peaks.

			The Collatz iteration starting at 7 is (7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which has 3 increasing peaks: 22, 34, and 52. No number smaller than 7 has 3 increasing peaks. Hence, a(3) = 7.

T. D. Noe, Table of n, a(n) for n = 0..60

Cf. A070165 (Collatz trajectory of n), A221469.

a221470 = (+ 1 ) . fromJust . (`elemIndex` (map a221469 [1..]))
-- Reinhard Zumkeller, Jan 18 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 20; t = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++; c = Collatz[n]; cnt = 0; mx = n; Do[If[k > mx, cnt++; mx = k], {k, c}]; If[cnt > 0 && cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; Join[{1}, t]

A346965 a(n) is the number of ascending subsequences in reducing n to 1 using the Collatz reduction, or -1 if n refutes the Collatz conjecture.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 0, 4, 1, 3, 1, 2, 3, 2, 0, 3, 4, 4, 1, 1, 3, 2, 1, 5, 2, 17, 3, 4, 2, 16, 0, 6, 3, 2, 4, 4, 4, 6, 1, 17, 1, 6, 3, 4, 2, 16, 1, 5, 5, 5, 2, 2, 17, 17, 3, 7, 4, 6, 2, 3, 16, 15, 0, 6, 6, 5, 3, 3, 2, 16, 4, 18, 4, 2, 4, 5, 6, 6, 1, 4, 17, 17

Offset: 1

Douglas Boffey, Aug 09 2021

nonn

In this sequence, a subsequence is considered ascending for as long as a (3*n + 1) / 2 step is required.

			a(9) = 4, viz.
  9->14;
  14->7->11->17->26;
  26->13->20;
  20->10->5->8.

Douglas Boffey, Table of n, a(n) for n = 1..20000
Douglas Boffey, Code used for generating b-file

Cf. A000265, A070168, A078719, A085062, A160541, A221469.

/* A007814 */
int num_clear_bits(unsigned n) {
  if (n == 0)
    return -1;
  return log2(n & -n);
}
int A346965(unsigned n) {
  int x;
  int result = 0;
  n >>= num_clear_bits(n);
  while (n > 1) {
    x = num_clear_bits(n + 1);
    n = ((n >> x) + 1) * pow(3, x) - 1;
    n >>= num_clear_bits(n);
    ++result;
  }
  return result;
}

a(2^n) = 0.
a((2^n*(2*x+1)-1) * 2^y) = a(3^n*(2*x+1)-1) + 1, where x, y >= 0.
a(n) = a(A085062(n)) + (n mod 2) for n >= 2. - Alan Michael Gómez Calderón, Feb 09 2025
a(n) = A160541(A000265(n)). - Alan Michael Gómez Calderón, Mar 19 2025

A350369 a(n) is the length of the longest sequence of consecutive tripling steps in the Collatz (3x+1) sequence beginning at n.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 3, 0, 3, 1, 2, 2, 1, 3, 4, 0, 1, 3, 2, 1, 1, 2, 3, 2, 2, 1, 6, 3, 2, 4, 6, 0, 2, 1, 2, 3, 3, 2, 3, 1, 6, 1, 3, 2, 1, 3, 6, 2, 3, 2, 2, 1, 1, 6, 6, 3, 3, 2, 2, 4, 3, 6, 6, 0, 3, 2, 2, 1, 1, 2, 6, 3, 6, 3, 2, 2, 2, 3, 4, 1, 3, 6, 6, 1, 1, 3, 3

Offset: 1

Kevin P. Thompson, Dec 27 2021

"Consecutive tripling steps" are repeated (3x+1)/2 operations that are not interrupted by a second division by 2.
This sequence attempts to measure the largest upward thrust in each Collatz sequence and so is correlated to some degree with the maximum value (A025586) and length (A006577) of Collatz sequences.
If n = 2^x * (2^y*z - 1), then a(n) >= y. - Charles R Greathouse IV, Oct 25 2022

			The Collatz sequence for n=7 has a streak of 3 consecutive tripling steps (at 7, 11, and 17), so a(7) = 3.
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
^      ^       ^

Kevin P. Thompson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A006577, A006667, A025586, A213215, A221469, A347270, A347409.

a(n)=my(c,r); n>>=valuation(n,2); while(n>1, n+=(n+1)/2; if(n%2, c++, r=max(r,c+1); n>>=valuation(n,2); c=0)); max(r,c) \\ Charles R Greathouse IV, Oct 25 2022

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A221470 Least m such that the Collatz (3x+1) iteration of m has exactly n increasing peak values.

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A346965 a(n) is the number of ascending subsequences in reducing n to 1 using the Collatz reduction, or -1 if n refutes the Collatz conjecture.

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A350369 a(n) is the length of the longest sequence of consecutive tripling steps in the Collatz (3x+1) sequence beginning at n.

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