A331460 Irregular triangle read by rows in which row n is the result of iterating the operation f(n) = n/7 if n == 0 (mod 7), otherwise f(n) = 7*(n + ceiling(n/7)), terminating at the first occurrence of 1.

1, 2, 21, 3, 28, 4, 35, 5, 42, 6, 49, 7, 1, 3, 28, 4, 35, 5, 42, 6, 49, 7, 1, 4, 35, 5, 42, 6, 49, 7, 1, 5, 42, 6, 49, 7, 1, 6, 49, 7, 1, 7, 1, 8, 70, 10, 84, 12, 98, 14, 2, 21, 3, 28, 4, 35, 5, 42, 6, 49, 7, 1, 9, 77, 11, 91, 13, 105, 15, 126, 18

Offset

1,2

Comments

f(n) is the operation C(n,m) = n/m if n == 0 (mod m) and m*(n + ceiling(n/m)) otherwise, where m = 7. The operations in the Collatz (3x + 1) problem (A070165), A330073, and A329263 are C(n,2), C(n,5), and C(n,8) respectively.

Conjecture: For any number n >= 1, there exists a k such that f^{k}(n) = 1, where f^{0}(n) = n and f^{k + 1}(n) = f(f^{k}(n)).

For any numbers n and k such that f^{k}(n) = 1, f^{k + 1}(7*n) = 1 and if n == 0 (mod 7) and n !== 0 or 7 (mod 56), then f^{k + 1}(floor(n/8)) = 1.

Links

Table of n, a(n) for n=1..71.

T. Ahmed and H. Snevily, Are There an Infinite Number of Collatz Integers?, 2013.

M. Bruschi, A generalization of the Collatz problem and conjecture, arXiv:0810.5169 [math.NT], 2008.

W. Carnielli, Some Natural Generalizations Of The Collatz Problem, Applied Mathematics E-Notes, 15 (2015), 197-206.

Formula

T(n,0) = n and T(n,k + 1) = T(n,k)/7 if T(n,k) == 0 (mod 7), 7*(T(n,k) + ceiling(T(n,k)/7)) otherwise, for n >= 1.

Example

The irregular array T(n,k) starts:
n\k   0   1   2   3   4    5   6    7   8    9  10  11  12  13  14  15  16 ...
1:    1
2:    2  21   3  28   4   35   5   42   6   49   7   1
3:    3  28   4  35   5   42   6   49   7    1
4:    4  35   5  42   6   49   7    1
5:    5  42   6  49   7    1
6:    6  49   7   1
7:    7   1
8:    8  70  10  84  12   98  14    2  21    3  28   4  35   5  42   6  49 ...
9:    9  77  11  91  13  105  15  126  18  147  21   3  28   4  35   5  42 ...
10:  10  84  12  98  14    2  21    3  28    4  35   5  42   6  49   7   1
...
T(8,18) = 1 and T(9,20) = 1.

Mathematica

f[n_] := NestWhileList[If[Mod[#, 7] == 0, #/7, 7 (Floor[#/7] + # + 1)] &, n, # > 1 &]; Flatten[Table[f[n], {n, 10}]]

Prog

(PARI) row(n)=my(N=List([n])); while(n>1, listput(N, n=if(n%7, 7*(n+ceil(n/7)), n/7))); Vec(N)

Crossrefs

Cf. A070165, A329263, A330073.

Sequence in context: A143247 A342079 A303216 * A303218 A162536 A377413

Adjacent sequences: A331457 A331458 A331459 * A331461 A331462 A331463

Keyword

nonn,easy,tabf

Author

Davis Smith, Jan 23 2020

Status

approved