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%I #26 May 06 2024 14:33:54
%S 1,1,3,1,5,5,1,1,1,7,1,1,1,11,9,1,1,1,17,7,11,1,1,1,13,11,17,13,1,1,1,
%T 5,17,13,5,15,1,1,1,1,13,5,1,23,17,1,1,1,1,5,1,1,35,13,19,1,1,1,1,1,1,
%U 1,53,5,29,21,1,1,1,1,1,1,1,5,1,11,1,23,1,1,1,1,1,1,1,1,1,17,1,35,25
%N Array read by upward antidiagonals: A(n, k) = R(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1, where Reduced Collatz function R(n) gives the odd part of 3n+1.
%C Collatz conjecture is equal to the claim that in each column 1 will eventually appear. See also arrays A372287 and A372288.
%H Antti Karttunen, <a href="/A372283/b372283.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals, flattened</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F For n > 1, A(n, k) = R(A372282(n-1, k)), where R(n) = (3*n+1)/2^A371093(n).
%F For all k >= 1, A(A258145(k-1), k) = 1 [which is the topmost 1 in each column].
%e Array begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%e ---+-----------------------------------------------------------------------
%e 1 | 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35,
%e 2 | 1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53,
%e 3 | 1, 1, 1, 17, 11, 13, 1, 35, 5, 11, 1, 53, 29, 31, 17, 71, 19, 5,
%e 4 | 1, 1, 1, 13, 17, 5, 1, 53, 1, 17, 1, 5, 11, 47, 13, 107, 29, 1,
%e 5 | 1, 1, 1, 5, 13, 1, 1, 5, 1, 13, 1, 1, 17, 71, 5, 161, 11, 1,
%e 6 | 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 13, 107, 1, 121, 17, 1,
%e 7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 161, 1, 91, 13, 1,
%e 8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 121, 1, 137, 5, 1,
%e 9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 91, 1, 103, 1, 1,
%e 10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 137, 1, 155, 1, 1,
%e 11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 103, 1, 233, 1, 1,
%e 12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 155, 1, 175, 1, 1,
%e 13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 233, 1, 263, 1, 1,
%e 14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 175, 1, 395, 1, 1,
%e 15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 263, 1, 593, 1, 1,
%e 16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 395, 1, 445, 1, 1,
%t With[{dmax = 15}, Table[#[[k, n-k+1]], {n, dmax}, {k, n}] & [Array[NestList[(3*# + 1)/2^IntegerExponent[3*# + 1, 2] &, 2*# - 1, dmax - #] &, dmax]]] (* _Paolo Xausa_, Apr 29 2024 *)
%o (PARI)
%o up_to = 91;
%o R(n) = { n = 1+3*n; n>>valuation(n, 2); };
%o A372283sq(n,k) = if(1==n,2*k-1,R(A372283sq(n-1,k)));
%o A372283list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372283sq((a-(col-1)),col))); (v); };
%o v372283 = A372283list(up_to);
%o A372283(n) = v372283[n];
%Y Cf. A005408 (row 1), A075677 (row 2), A372443 (column 14).
%Y Cf. A258145 (A075680), A371093.
%Y Arrays derived from this one or related to:
%Y A372282,
%Y A372287 the column index of A(n, k) in array A257852,
%Y A372361 terms xored with binary words of the same length, either of the form 10101...0101 or 110101...0101, depending on whether the binary length is odd or even,
%Y A372360 binary weights of A372361.
%Y Cf. also array A371095 (giving every fourth column, 1, 5, 9, ...) and irregular array A256598 which gives the terms of each column, but only down to the first 1.
%K nonn,tabl
%O 1,3
%A _Antti Karttunen_, Apr 28 2024
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