A372288 Array read by upward antidiagonals: A(n, k) = A265745(A372282(n, k)), n,k >= 1, where A265745(n) is the sum of digits in "Jacobsthal greedy base".

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 5, 3, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 5, 3

Offset

1,10

Comments

Collatz conjecture is equal to the claim that each column will eventually settle to constant 1's, somewhere under its topmost row. This works as only the bisection A002450 of Jacobsthal numbers (A001045) contains numbers of the form 4k+1, while the other bisection contains only numbers of the form 4k+3, which do not occur among the range of A372351. See also the comments in A371094.

Links

Table of n, a(n) for n=1..105.

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13     14 15    16 17 18 19 20    21 22
---+----------------------------------------------------------------------------
1  | 1, 1, 1, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3,     3, 3,    3, 3, 3, 3, 5,    5, 1,
2  | 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 5, 5,     5, 3,    5, 3, 3, 3, 5,    5, 3,
3  | 1, 1, 1, 3, 3, 3, 1, 5, 1, 3, 1, 3, 3,     5, 3,    5, 5, 1, 3, 3,    5, 3,
4  | 1, 1, 1, 3, 3, 1, 1, 3, 1, 3, 1, 1, 3,     5, 3,    3, 3, 1, 3, 5,    5, 3,
5  | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3,     5, 1,    5, 3, 1, 3, 3,    3, 3,
6  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3,     3, 1,    5, 3, 1, 1, 5,    5, 3,
7  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    3, 3, 1, 1, 3,    5, 3,
8  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 3,    3, 3,
9  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     3, 1,    5, 1, 1, 1, 3,    5, 1,
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 3,    5, 1,
11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1, 2155, 1, 1, 1, 1,    5, 1,
12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 1, 6251, 1,
13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10347, 1,    5, 1, 1, 1, 1,    5, 1,
14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 1,    5, 1,
15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    7, 1, 1, 1, 1,    5, 1,
16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,     5, 1,    5, 1, 1, 1, 1,    7, 1,

Prog

(PARI)
up_to = 105;
A130249(n) = (#binary(3*n+1)-1);
A001045(n) = (2^n - (-1)^n) / 3;
A265745(n) = { my(s=0); while(n, s++; n -= A001045(A130249(n))); (s); };
A371094(n) = { my(m=1+3*n, e=valuation(m, 2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n, k) = if(1==n, 2*k-1, A371094(A372282sq(n-1, k)));
A372288sq(n, k) = A265745(A372282sq(n, k));
A372288list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A372288sq((a-(col-1)), col))); (v); };
v372288 = A372288list(up_to);
A372288(n) = v372288[n];

Crossrefs

Cf. A001045, A002450, A265745, A265747, A371094, A372351, A372282, A372283, A372287.

Cf. also array A372561 (formed by columns whose indices in this array are given by A372443).

Sequence in context: A295931 A295920 A176187 * A180683 A375360 A365331

Adjacent sequences: A372285 A372286 A372287 * A372289 A372290 A372291

Keyword

nonn,tabl

Author

Antti Karttunen, Apr 28 2024

Status

approved