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%I A372283 #26 May 06 2024 14:33:54
%S A372283 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 A372283 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 A372283 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 A372283 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 A372283 Collatz conjecture is equal to the claim that in each column 1 will eventually appear. See also arrays A372287 and A372288.
%H A372283 Antti Karttunen, <a href="/A372283/b372283.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals, flattened</a>
%H A372283 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A372283 For n > 1, A(n, k) = R(A372282(n-1, k)), where R(n) = (3*n+1)/2^A371093(n).
%F A372283 For all k >= 1, A(A258145(k-1), k) = 1 [which is the topmost 1 in each column].
%e A372283 Array begins:
%e A372283 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%e A372283 ---+-----------------------------------------------------------------------
%e A372283 1 | 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35,
%e A372283 2 | 1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53,
%e A372283 3 | 1, 1, 1, 17, 11, 13, 1, 35, 5, 11, 1, 53, 29, 31, 17, 71, 19, 5,
%e A372283 4 | 1, 1, 1, 13, 17, 5, 1, 53, 1, 17, 1, 5, 11, 47, 13, 107, 29, 1,
%e A372283 5 | 1, 1, 1, 5, 13, 1, 1, 5, 1, 13, 1, 1, 17, 71, 5, 161, 11, 1,
%e A372283 6 | 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 13, 107, 1, 121, 17, 1,
%e A372283 7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 161, 1, 91, 13, 1,
%e A372283 8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 121, 1, 137, 5, 1,
%e A372283 9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 91, 1, 103, 1, 1,
%e A372283 10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 137, 1, 155, 1, 1,
%e A372283 11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 103, 1, 233, 1, 1,
%e A372283 12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 155, 1, 175, 1, 1,
%e A372283 13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 233, 1, 263, 1, 1,
%e A372283 14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 175, 1, 395, 1, 1,
%e A372283 15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 263, 1, 593, 1, 1,
%e A372283 16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 395, 1, 445, 1, 1,
%t A372283 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 A372283 (PARI)
%o A372283 up_to = 91;
%o A372283 R(n) = { n = 1+3*n; n>>valuation(n, 2); };
%o A372283 A372283sq(n,k) = if(1==n,2*k-1,R(A372283sq(n-1,k)));
%o A372283 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 A372283 v372283 = A372283list(up_to);
%o A372283 A372283(n) = v372283[n];
%Y A372283 Cf. A005408 (row 1), A075677 (row 2), A372443 (column 14).
%Y A372283 Cf. A258145 (A075680), A371093.
%Y A372283 Arrays derived from this one or related to:
%Y A372283 A372282,
%Y A372283 A372287 the column index of A(n, k) in array A257852,
%Y A372283 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 A372283 A372360 binary weights of A372361.
%Y A372283 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 A372283 nonn,tabl
%O A372283 1,3
%A A372283 _Antti Karttunen_, Apr 28 2024