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A372283
Array read by upward antidiagonals: A(n, k) = R(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1, where Reduced Collatz function R(n) gives the odd part of 3n+1.
10
1, 1, 3, 1, 5, 5, 1, 1, 1, 7, 1, 1, 1, 11, 9, 1, 1, 1, 17, 7, 11, 1, 1, 1, 13, 11, 17, 13, 1, 1, 1, 5, 17, 13, 5, 15, 1, 1, 1, 1, 13, 5, 1, 23, 17, 1, 1, 1, 1, 5, 1, 1, 35, 13, 19, 1, 1, 1, 1, 1, 1, 1, 53, 5, 29, 21, 1, 1, 1, 1, 1, 1, 1, 5, 1, 11, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 35, 25
OFFSET
1,3
COMMENTS
Collatz conjecture is equal to the claim that in each column 1 will eventually appear. See also arrays A372287 and A372288.
FORMULA
For n > 1, A(n, k) = R(A372282(n-1, k)), where R(n) = (3*n+1)/2^A371093(n).
For all k >= 1, A(A258145(k-1), k) = 1 [which is the topmost 1 in each column].
EXAMPLE
Array begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
---+-----------------------------------------------------------------------
1 | 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35,
2 | 1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53,
3 | 1, 1, 1, 17, 11, 13, 1, 35, 5, 11, 1, 53, 29, 31, 17, 71, 19, 5,
4 | 1, 1, 1, 13, 17, 5, 1, 53, 1, 17, 1, 5, 11, 47, 13, 107, 29, 1,
5 | 1, 1, 1, 5, 13, 1, 1, 5, 1, 13, 1, 1, 17, 71, 5, 161, 11, 1,
6 | 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 13, 107, 1, 121, 17, 1,
7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 161, 1, 91, 13, 1,
8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 121, 1, 137, 5, 1,
9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 91, 1, 103, 1, 1,
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 137, 1, 155, 1, 1,
11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 103, 1, 233, 1, 1,
12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 155, 1, 175, 1, 1,
13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 233, 1, 263, 1, 1,
14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 175, 1, 395, 1, 1,
15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 263, 1, 593, 1, 1,
16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 395, 1, 445, 1, 1,
MATHEMATICA
With[{dmax = 15}, Table[#[[k, n-k+1]], {n, dmax}, {k, n}] & [Array[NestList[(3*# + 1)/2^IntegerExponent[3*# + 1, 2] &, 2*# - 1, dmax - #] &, dmax]]] (* Paolo Xausa, Apr 29 2024 *)
PROG
(PARI)
up_to = 91;
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372283sq(n, k) = if(1==n, 2*k-1, R(A372283sq(n-1, k)));
A372283list(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] = A372283sq((a-(col-1)), col))); (v); };
v372283 = A372283list(up_to);
A372283(n) = v372283[n];
CROSSREFS
Cf. A005408 (row 1), A075677 (row 2), A372443 (column 14).
Arrays derived from this one or related to:
A372287 the column index of A(n, k) in array A257852,
A372361 terms xored with binary words of the same length, either of the form 10101...0101 or 110101...0101, depending on whether the binary length is odd or even,
A372360 binary weights of A372361.
Cf. also array A371095 (giving every fourth column, 1, 5, 9, ...) and irregular array A256598 which gives the terms of each column, but only down to the first 1.
Sequence in context: A094648 A096975 A145174 * A351965 A228785 A135184
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Apr 28 2024
STATUS
approved