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 A343858 #69 Jul 12 2021 03:10:37 %S A343858 0,0,1,0,1,1,0,1,1,3,0,1,1,1,1,0,1,1,3,1,5,0,1,1,3,1,5,3,0,1,1,3,1,5, %T A343858 3,7,0,1,1,3,1,5,3,7,1,0,1,1,3,1,1,3,7,1,9,0,1,1,3,1,5,3,7,1,3,5,0,1, %U A343858 1,3,1,5,3,7,1,9,5,5,0,1,1,3,1,5,3,7,1,9,5,1 %N A343858 Square array T(m,n), read by ascending antidiagonals. Let f(k) = k/2 if k is even, otherwise ((2*n+1)*k+2*r+1)/2, r is the smallest integer greater than -1, where m = f^j(m) for j > 0 exists and is determined in A345228, T(m,n) is the smallest number reached in the cyclic trajectory of m = f^j(m). f^j(m) means j times recursion into f(m). %C A343858 This sequence, together with A345228, provides information regarding generalized Collatz functions. (Replace 3*k+1 in the standard Collatz function with a more general a*k+b; then a = 1+2*n and b = 1+2*A345228(m,n).) A345228 tells us which m are part of a cyclic orbit but not if these are part of the same cycle. This sequence identifies each distinct cycle with a different number. Example: If A345228(m1,n) = A345228(m2,n) we know m1 and m2 are part of a cycle but not necessarily the same cycle. If T(m1,n) <> T(m2,n) we know m1 and m2 are not in the same cycle. %C A343858 The value of n appears to have only a small effect in this sequence and in a majority of cases we find T(m,n) = A000265(m) holds true. This is surprising, given how n is involved in the definition. %H A343858 Thomas Scheuerle, <a href="/A343858/a343858.svg">Yellow dot if T(m,n) <> A000265(n); m = 1..100; n = 1..20</a>. %H A343858 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>. %F A343858 T((1+2*n)*m,n)/T(m,n) = 1+2*n. %F A343858 T((1+2*(n-b))*m,n)/T(m,n) = 1+2*(n-b). 0 <= b <= n. This formula is only for the majority of cases true if b > 0. For each column m are some rows n where an exception will be seen. %F A343858 T(m,n) <= A000265(m) (largest odd divisor of m). %F A343858 T(m,n) = A000265(m) For the majority of all n. %e A343858 Twelve initial terms of rows 0-10 are listed below: %e A343858 n |m-> %e A343858 0: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 1, 11, ... %e A343858 1: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, ... %e A343858 2: 0, 1, 1, 1, 1, 5, 3, 7, 1, 3, 5, 1, ... %e A343858 3: 0, 1, 1, 3, 1 5, 3, 7, 1, 9, 5, 11, ... %e A343858 4: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... %e A343858 5: 0, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 11, ... %e A343858 6: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 1, ... %e A343858 7: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... %e A343858 8: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... %e A343858 9: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... %e A343858 10: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ... %e A343858 Example: T(3,4) = 3 -> f(n): k/2; (9*k+21)/2. This is because r = A345228(3,4) = 10 and 2*10+1 = 21. %e A343858 f(3) = 24, f(24) = 12, f(12) = 6, f(6) = 3, f(3) = 24, .... %e A343858 The smallest number in this cycle is 3. %Y A343858 Cf. A344583, A345228, A000265. %K A343858 nonn,tabl %O A343858 0,10 %A A343858 _Thomas Scheuerle_, Jun 14 2021