A096773 -id:A096773 - cp's OEIS frontend

A372352 The difference between n and the largest term of A086893 <= n.

0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6

Offset: 1

Antti Karttunen (proposed by Ali Sada), Apr 29 2024

The terms a(n) grow from 0 (whenever n is in A086893) by 1 until the next element of A086893 is reached. - M. F. Hasler, May 08 2025

Cf. A086893, A096773, A372353.

Cf. also A372286.

A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372352(n) = { my(k); for(i=1,oo,k=A086893(i); if(k>n, return(n-A086893(i-1)))); };

a(n) = n - max ( A086893 intersect [1..n] ) (= 0 iff n in A086893). - M. F. Hasler, May 08 2025

A329480 a(n) = (1 - A075677(n))/6 if 6|(A075677(n)-1) or a(n) = (A075677(n) + 1)/6 if 6|(A075677(n)+1).

Original entry on oeis.org

0, 1, 0, 2, -1, 3, 1, 4, -2, 5, 0, 6, -3, 7, 2, 8, -4, 9, -1, 10, -5, 11, 3, 12, -6, 13, 1, 14, -7, 15, 4, 16, -8, 17, -2, 18, -9, 19, 5, 20, -10, 21, 0, 22, -11, 23, 6, 24, -12, 25, -3, 26, -13, 27, 7, 28, -14, 29, 2, 30, -15, 31, 8, 32, -16, 33, -4, 34, -17

Offset: 1

Fabian S. Reid, Jun 07 2020

sign

A fractal sequence.
This sequence is related to the Collatz Problem and can be illustrated on a logarithmic spiral to determine the odd numbers in the trajectory of a natural number of the form 6x+1 or 6x-1 simply by moving forward if the integer is positive, backward if the integer is negative, and continuing this forward-backward movement indefinitely.
When formatted as a table T with 4 columns, the third column T(n,3) is equal to the sequence. - Ruud H.G. van Tol, Oct 16 2023

			For n = 2, A075677(2) = 5, so a(2) = 1.
For n = 9, A075677(9) = 13, so a(9) = -2.
From _Ruud H.G. van Tol_, Oct 16 2023: (Start)
Array T begins:
 n|k_1|__2|__3|__4|
 1|  0   1   0   2
 2| -1   3   1   4
 3| -2   5   0   6
 4| -3   7   2   8
 5| -4   9  -1  10
 6| -5  11   3  12
... (End)

Ruud H.G. van Tol, Table of n, a(n) for n = 1..10000
Commons.Wikimedia Integer Spiral
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.

Cf. A002450, A016789, A075677, A096773, A349414.

nterms=100;Table[r=(c=3(2n-1)+1)/2^IntegerExponent[c,2];If[Mod[r,6]==1,(1-r)/6,(1+r)/6],{n, nterms}] (* Paolo Xausa, Nov 28 2021 *)

a(n) = my(x=3*n-1); x>>=valuation(x, 2); if(1==x%6, 1-x, 1+x)/6; \\ Ruud H.G. van Tol, Oct 16 2023

a(n) = (1 - A075677(n))/6 when 1 = A075677(n) mod 6, or
a(n) = (A075677(n) + 1)/6 when 5 = A075677(n) mod 6.
From Ruud H.G. van Tol, Oct 16 2023: (Start)
a(4*n-1) = a(n).
T(n,1) = 1-n; T(n,2) = 2*n-1 = n - T(n,1); T(n,3) = T(floor((n-1)/4) + 1, (n-1) mod 4 + 1) = a(n); T(n,4) = 2*n = T(n,2) + 1. (End)

A375961 2-adic valuation of 6*n + 2.

Original entry on oeis.org

1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 9, 1

Offset: 0

Ruud H.G. van Tol, Sep 04 2024

6*i+2 is the first (3*x+1)/2 successor of 4*i+1, with i >= 0.

The first occurrence of odd t is before that of t-1.

			a(21) = A007814(6*21+2) = 7.

Ruud H.G. van Tol, Table of n, a(n) for n = 0..10000
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A007814, A016933, A087229, A087230, A096773, A371093, A375782.

a[n_] := IntegerExponent[6*n + 2, 2]; Array[a, 100, 0] (* Amiram Eldar, Sep 04 2024 *)

PARI
```
a(n) = valuation(6*n+2, 2);
```

def A375961(n): return (~(3*n+1)&3*n).bit_length()+1 # Chai Wah Wu, Sep 27 2024

a(n) = A007814(6*n + 2).
a(n) = A371093(n) + 1.
a(n) = A087229(n) - 1.
a(n) = k for n == A096773(k) (mod 2^k), k >= 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Oct 01 2024

A364602 Triangle T(n,k) with rows of length 2n-1, generated by T(1,1)=0, T(n,1)=T(n-1,1)+2, T(n,2)=4(n-1)-1, and for k>=3, T(n,k)=4*T(n-1,k-2)+1.

Original entry on oeis.org

0, 2, 3, 1, 4, 7, 9, 13, 5, 6, 11, 17, 29, 37, 53, 21, 8, 15, 25, 45, 69, 117, 149, 213, 85, 10, 19, 33, 61, 101, 181, 277, 469, 597, 853, 341, 12, 23, 41, 77, 133, 245, 405, 725, 1109, 1877, 2389, 3413, 1365, 14, 27, 49, 93, 165, 309, 533, 981, 1621, 2901

Offset: 1

Ruud H.G. van Tol, Jul 29 2023

The sequence is a permutation of all integers >= 0.
Each row of T contains n*2-1 terms; the terms in column k increase by 2^k.
T(1,1) = 0; T(2,2) = 3.
T(2,1) = T(1,1)+2 = 2; T(2,3) = 4*T(1,1)+1 = 1 ("knight jump").
In the context of the 3x+1 problem, when a term x is used to represent the odd 4*x+1, its successor is 3*x+1, and k-1 is the 2-adic valuation of 3*x+1.
Right diagonal is A002450.
The terms at the top of the columns are A096773(k), or (2^(k-1)*(3 + 2*(-1)^k) - 1)/3.
When the table is analytically continued upwards by subtracting 2^k, the first layer of values are -A255138(k), or -(2^k*(3 + 2*(-1)^k) + 1)/3.

			Triangle T(n,k) begins:
n/k 1| 2| 3| 4|  5|  6|  7|  8|  9| 10| 11|
1|  0
2|  2  3  1
3|  4  7  9 13   5
4|  6 11 17 29  37  53  21
5|  8 15 25 45  69 117 149 213  85
6| 10 19 33 61 101 181 277 469 597 853 341
7| 12 ...

Cf. A002450, A007310, A071797, A087445, A096773, A255138, A257480.

my(N=8, v=Vec([0, 2, 3, 1], N^2), p=4); for(n=3, N, my(K=2*n-1); for(k=1, K, v[p+k]=if(k<=2, v[p-K+k+2]+2^k, 4*v[p-K+k]+1)); p+=K); v

T(n, k) = 2^k*(n-(6*k+3-(-1)^k)/12)-1/3;

n_of_x(x) = my(n=0); while(1==x%4, x>>=2; n++); n + if(x%2,(x+1)/4,  x/2) + 1;

PARI
```
k_of_x(x) = valuation(3*x+1,2) + 1;
```

For n>1, T(n,k) = T(n-1,k) + 2^k, so T(n,1) = 2*(n-1).
T(n,2) = 4*(n-1)-1 = 2*T(n,1)-1, so T(2,2) = 3.
For n>1 and k>2, T(n,k) = 4*T(n-1,k-2)+1, so T(2,3) = 1.
For i>=0, a(i^2+1) = T(i+1,1).
T(n, k) = 2^k * (n - (6*k + 3 - (-1)^k)/12) - 1/3.
T(n,1) == 0 (mod 2); T(n,2) == 3 (mod 4); T(n,k>=3) == 1 (mod 4).
k = v2(3*T(n,k)+1) + 1, where v2(x) = A007814(x) is the 2-adic valuation of x.

cp's OEIS Frontend

A372352 The difference between n and the largest term of A086893 <= n.

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A329480 a(n) = (1 - A075677(n))/6 if 6|(A075677(n)-1) or a(n) = (A075677(n) + 1)/6 if 6|(A075677(n)+1).

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A375961 2-adic valuation of 6*n + 2.

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Mathematica

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Formula

A364602 Triangle T(n,k) with rows of length 2*n-1, generated by T(1,1)=0, T(n,1)=T(n-1,1)+2, T(n,2)=4*(n-1)-1, and for k>=3, T(n,k)=4*T(n-1,k-2)+1.

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PARI

PARI

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PARI

Formula

A364602 Triangle T(n,k) with rows of length 2n-1, generated by T(1,1)=0, T(n,1)=T(n-1,1)+2, T(n,2)=4(n-1)-1, and for k>=3, T(n,k)=4*T(n-1,k-2)+1.