This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A331460 #89 Aug 06 2024 11:13:56
%S A331460 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,
%T A331460 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,
%U A331460 5,42,6,49,7,1,9,77,11,91,13,105,15,126,18
%N 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.
%C A331460 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.
%C A331460 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)).
%C A331460 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.
%H A331460 T. Ahmed and H. Snevily, <a href="https://citeseerx.ist.psu.edu/pdf/2207919fcc0e2e4336faf345389c5dcd2daea50e">Are There an Infinite Number of Collatz Integers?</a>, 2013.
%H A331460 M. Bruschi, <a href="https://arxiv.org/abs/0810.5169">A generalization of the Collatz problem and conjecture</a>, arXiv:0810.5169 [math.NT], 2008.
%H A331460 W. Carnielli, <a href="https://www.emis.de/journals/AMEN/2015/AMEN(150711).pdf">Some Natural Generalizations Of The Collatz Problem</a>, Applied Mathematics E-Notes, 15 (2015), 197-206.
%F A331460 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.
%e A331460 The irregular array T(n,k) starts:
%e A331460 n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
%e A331460 1: 1
%e A331460 2: 2 21 3 28 4 35 5 42 6 49 7 1
%e A331460 3: 3 28 4 35 5 42 6 49 7 1
%e A331460 4: 4 35 5 42 6 49 7 1
%e A331460 5: 5 42 6 49 7 1
%e A331460 6: 6 49 7 1
%e A331460 7: 7 1
%e A331460 8: 8 70 10 84 12 98 14 2 21 3 28 4 35 5 42 6 49 ...
%e A331460 9: 9 77 11 91 13 105 15 126 18 147 21 3 28 4 35 5 42 ...
%e A331460 10: 10 84 12 98 14 2 21 3 28 4 35 5 42 6 49 7 1
%e A331460 ...
%e A331460 T(8,18) = 1 and T(9,20) = 1.
%t A331460 f[n_] := NestWhileList[If[Mod[#, 7] == 0, #/7, 7 (Floor[#/7] + # + 1)] &, n, # > 1 &]; Flatten[Table[f[n], {n, 10}]]
%o A331460 (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)
%Y A331460 Cf. A070165, A329263, A330073.
%K A331460 nonn,easy,tabf
%O A331460 1,2
%A A331460 _Davis Smith_, Jan 23 2020