A330073 -id:A330073 - cp's OEIS frontend

A331272 Irregular triangle in which row n lists numbers m such that A330073(m,n) = 1.

1, 5, 25, 4, 125, 20, 625, 3, 100, 104, 3125, 15, 16, 500, 520, 15625, 2, 75, 80, 83, 86, 2500, 2600, 2604, 78125, 10, 12, 13, 375, 400, 415, 416, 430, 433, 12500, 13000, 13020, 390625, 50, 60, 62, 65, 66, 69, 71, 1875, 2000, 2075, 2080, 2083, 2150, 2165, 2166, 62500, 65000, 65100, 65104, 1953125

Offset: 0

Davis Smith, Jan 13 2020

The last number in row n is 5^n (A000351(n)).
For numbers m and n such that m is in row n, 5*m is in row n + 1 and if m > 10 and m == 10, 15, 20, or 25 (mod 30), then floor(m/6) is in row n + 1.
The conjecture in A330073 claims that every positive integer appears in this triangle.

			The irregular triangle starts:
0:   1
1:   5
2:  25
3:   4  125
4:  20  625
5:   3  100  104 3125
6:  15   16  500  520 15625
7:   2   75   80   83    86  2500  2600  2604 78125
8:  10   12   13  375   400   415   416   430   433 12500 13000 13020 390625

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.

Cf. A000351 (5^n), A127824, A330073.

A331272(lim)=my(N=[1], b=-1, RC=5*[2..5]); while(bif(setsearch(RC,X%30)&&(X>RC[1]),[floor(X/6),5*X],X*5),N))[1,]))

If N is the list of numbers in row n, then the list of numbers in row n + 1 is the union of each number in N multiplied by 5 and numbers floor(x/6) where x is in N, congruent to 0 (mod 5), not congruent to 0 or 5 (mod 30), and floor(x/6) > 1.

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.

Original entry on oeis.org

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

Davis Smith, Jan 23 2020

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.

			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.

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.

Cf. A070165, A329263, A330073.

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

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

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.

cp's OEIS Frontend

A331272 Irregular triangle in which row n lists numbers m such that A330073(m,n) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula