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A246646 Irregular triangle T(n,m) with sieved modified Collatz sequences for k = A246647(n), n >= 1, m = 1, ..., A248154(n).

%I A246646 #15 Feb 16 2025 08:33:23
%S A246646 2,1,3,5,8,4,2,6,3,7,11,17,26,13,20,10,5,9,14,7,12,6,15,23,35,53,80,
%T A246646 40,20,16,8,18,9,19,29,44,22,11,21,32,16,24,12,25,38,19,27,41,62,31,
%U A246646 47,71,107,161,242,121,182,91,137,206,103,155,233,350,175,263,395,593,890,445,668,334,167,251,377,566,283,425,638,319,479,719,1079,1619,2429,3644,1822,911,1367,2051,3077,4616,2308,1154,577,866,433,650,325,488,244,122,61,92,46,23
%N A246646 Irregular triangle T(n,m) with sieved modified Collatz sequences for k = A246647(n), n >= 1, m = 1, ..., A248154(n).
%C A246646 The row length sequence for this irregular triangle is A248154.
%C A246646 The (modified or Terras) Collatz map is T(k) = (3*k +1)/2 if k is odd and T(k) = k/2 if k is even. See the array A070168.
%C A246646 The present irregular array starts with row n=1 for k=2 with 2, 1 and ends because the next number would be 2 which appeared already in this row (this is the trivial cycle). Row n=2 for k=3 is then 3, 5, 8, 4, 2 and stops with 2 which is the first number in this row which appeared already in row k=1. A row for k=4 does not show up because 4 already appeared in the row for k=3. Also no row for k=5 appears. Row n=3 is for k=6 with 6,3, etc.
%C A246646 In this way a 'minimal' Collatz table is build. The Collatz conjecture is that every positive integer is present (the end numbers in each row n >= 2 appear exactly twice).
%H A246646 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>,
%e A246646 The irregular triangle T(n,m) begins:
%e A246646 n,   k \ m
%e A246646 1,   2:   2   1
%e A246646 2,   3:   3   5  8  4  2
%e A246646 3,   6:   6   3
%e A246646 4,   7:   7  11 17 26 13 20 10  5
%e A246646 5,   9:   9  14  7
%e A246646 6,  12:  12   6
%e A246646 7,  15:  15  23 35 53 80 40 20
%e A246646 8,  16:  16   8
%e A246646 9,  18:  18   9
%e A246646 10, 19:  19  29 44 22 11
%e A246646 11, 21:  21  32 16
%e A246646 12, 24:  24  12
%e A246646 13, 25:  25  38 19
%e A246646 ...
%e A246646 Row n=14, k=27: 27 41 62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23;
%e A246646 Row n=15, k=28: 28 14;
%e A246646 Row n=16, k=30: 30 15; ...
%e A246646 The complete modified Collatz iteration until 1 is reached is obtained, for example for k=19, as follows: 19  29 44 22 11, (11) 17 26 13 20 10 5, (5) 8 4 2, (2) 1, that is 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1, which is row n=19 of A070168.
%Y A246646 Cf. A246647, A248154, A070168.
%K A246646 nonn,tabf,easy
%O A246646 1,1
%A A246646 _Wolfdieter Lang_, Oct 02 2014