A330073 Irregular triangle read as rows in which row n is the result of iterating the operation f(n) = n/5 if n == 0 (mod 5), otherwise f(n) = 5*(floor(n/5) + n + 1), terminating at the first occurrence of 1.

1, 2, 15, 3, 20, 4, 25, 5, 1, 3, 20, 4, 25, 5, 1, 4, 25, 5, 1, 5, 1, 6, 40, 8, 50, 10, 2, 15, 3, 20, 4, 25, 5, 1, 7, 45, 9, 55, 11, 70, 14, 85, 17, 105, 21, 130, 26, 160, 32, 195, 39, 235, 47, 285, 57, 345, 69, 415, 83, 500, 100, 20, 4, 25, 5, 1, 8, 50

Offset: 1

Author

Davis Smith, Nov 30 2019

Keywords

• nonn
• easy
• tabf

Comments

f(n) is the operation C(n,m) = n/m if n == 0 (mod m), m*(floor(n/m) + n + 1) otherwise where m = 5. C(n,2) is the operation in the Collatz problem (A070165) and C(n,8) is the operation in A329263.

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

For any number n, if n is a power of 5 multiplied by 1, 2, 3, or 4, then there is a number k such that f^{k}(n) = 1. If n is congruent to 10, 15, 20, or 25 (mod 30) and there is a k such that f^{k}(n) = 1, then f^{k + 1}(floor(n/6)) = 1.

Examples

			The irregular array T(n,k) starts:
n\k   0   1   2   3   4   5   6   7   8   9   10   11   12   13 ...
1:    1
2:    2  15   3  20   4  25   5   1
3:    3  20   4  25   5   1
4:    4  25   5   1
5:    5   1
6:    6  40   8  50  10   2  15   3  20   4   25    5    1
7:    7  45   9  55  11  70  14  85  17 105   21  130   26  160 ...
8:    8  50  10   2  15   3  20   4  25   5    1
9:    9  55  11  70  14  85  17 105  21 130   26  160   32  195 ...
10:  10   2  15   3  20   4  25   5   1
T(7,31) = 1 and T(9,29) = 1.
		

Crossrefs

Cf. A070165, A329263.

Programs

Mathematica

Array[NestWhileList[If[Mod[#, 5] == 0, #/5, 5 (Floor[#/5] + # + 1)] &, #, # > 1 &] &, 8] // Flatten (* Michael De Vlieger, Dec 01 2019 *)

PARI

row(n)=my(N=[n],m=5);while(n>1,N=concat(N,n=if(n%m,m*(n+floor(n/m)+1),n/m)));N

Formula

T(n,0) = n, T(n,k + 1) = T(n,k)/5 if T(n,k) == 0 (mod 5), 5*(T(n,k) + floor(T(n,k)/5) + 1) otherwise, for n >= 1.