A160541 -id:A160541 - cp's OEIS frontend

A160542 Not divisible by 11.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112

Offset: 1

Zerinvary Lajos, May 18 2009

Contains numbers like 100, 111, 112, 113 which are not in A043096. - R. J. Mathar, May 20 2009

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A043096.

A160541 := proc(n)
    option remember ;
    if n <= 10 then
        n;
    else
        procname(n-10)+11 ;;
    end if;
end proc:
seq(A160541(n),n=1..100) ; # R. J. Mathar, Aug 05 2022

Select[Table[n,{n,200}],Mod[#,11]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{1,2,3,4,5,6,7,8,9,10,12},70] (* Harvey P. Dale, Sep 16 2020 *)

[i for i in range(72) if gcd(11, i) == 1]

a(n) = a(n-10) + 11, n>10. - R. J. Mathar, May 20 2009
G.f.: x*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10) / ( (1+x)*(1+x+x^2+x^3+x^4)*(x^4-x^3+x^2-x+1)*(x-1)^2 ). - R. J. Mathar, May 02 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = (cot(Pi/11) - cot(2*Pi/11) + tan(Pi/22) - tan(3*Pi/22) + tan(5*Pi/22)) * Pi/11. - Amiram Eldar, May 11 2025

A363270 The result, starting from n, of Collatz steps x -> (3x+1)/2 while odd, followed by x -> x/2 while even.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 13, 1, 7, 5, 13, 3, 5, 7, 5, 1, 13, 9, 11, 5, 1, 11, 5, 3, 19, 13, 31, 7, 11, 15, 121, 1, 25, 17, 5, 9, 7, 19, 67, 5, 31, 21, 49, 11, 17, 23, 121, 3, 37, 25, 29, 13, 5, 27, 47, 7, 43, 29, 67, 15, 23, 31, 91, 1, 49, 33, 19, 17, 13, 35, 121, 9, 55

Offset: 1

Dustin Theriault, May 23 2023

Each x -> (3x+1)/2 step decreases the number of trailing 1-bits by 1 so A007814(n+1) of them, and the result of those steps is 2*A085062(n).

Dustin Theriault, Table of n, a(n) for n = 1..10000

Cf. A000265, A085062.
Cf. A160541 (number of iterations).
Cf. A075677.

int a(int n) {
  while (n & 1) n += (n >> 1) + 1;
  while (!(n & 1)) n >>= 1;
  return n;
}

OddPart[x_] := x / 2^IntegerExponent[x, 2]
Table[OddPart[(3/2)^IntegerExponent[i + 1, 2] * (i + 1) - 1], {i, 100}]

oddpart(n) = n >> valuation(n, 2); \\ A000265
a(n) = oddpart((3/2)^valuation(n+1, 2)*(n+1) - 1); \\ Michel Marcus, May 24 2023

a(n) = OddPart((3/2)^A007814(n+1)*(n+1) - 1), where OddPart(t) = A000265(t).

a(n) = OddPart(A085062(n)).

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

cp's OEIS Frontend

A160542 Not divisible by 11.

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A363270 The result, starting from n, of Collatz steps x -> (3x+1)/2 while odd, followed by x -> x/2 while even.

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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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