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%I #9 Apr 28 2024 21:23:57
%S 1,1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,3,2,3,1,1,1,1,3,3,1,1,1,1,1,3,1,
%T 1,4,1,1,1,1,1,1,1,6,3,1,1,1,1,1,1,1,9,1,5,1,1,1,1,1,1,1,1,1,2,1,1,1,
%U 1,1,1,1,1,1,1,3,1,6,1,1,1,1,1,1,1,1,1,3,1,9,4,1,1,1,1,1,1,1,1,1,1,1,1,5,7
%N Array read by upward antidiagonals: A(n, k) = A371092(A372283(n, k)), n,k >= 1.
%C A(n, k) gives the column index of A372282(n, k) [or equally, of A372283(n, k)] in array A257852.
%C Collatz conjecture is equal to the claim that in each column 1 will eventually appear.
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A(n, k) = A371092(A372282(n,k)) = A371092(A372283(n,k)).
%e Array begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
%e ---+---------------------------------------------------------------
%e 1 | 1, 1, 1, 2, 2, 3, 1, 4, 3, 5, 1, 6, 4, 7, 2, 8, 5, 9, 2, 10,
%e 2 | 1, 1, 1, 3, 2, 3, 1, 6, 1, 2, 1, 9, 5, 6, 3, 12, 4, 1, 2, 15,
%e 3 | 1, 1, 1, 3, 3, 1, 1, 9, 1, 3, 1, 1, 2, 8, 3, 18, 5, 1, 3, 12,
%e 4 | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 12, 1, 27, 2, 1, 3, 17,
%e 5 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 18, 1, 21, 3, 1, 1, 4,
%e 6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 16, 3, 1, 1, 5,
%e 7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 1, 23, 1, 1, 1, 2,
%e 8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 18, 1, 1, 1, 3,
%e 9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 26, 1, 1, 1, 3,
%e 10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 18, 1, 39, 1, 1, 1, 1,
%e 11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 30, 1, 1, 1, 1,
%e 12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 44, 1, 1, 1, 1,
%e 13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 30, 1, 66, 1, 1, 1, 1,
%e 14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 44, 1, 99, 1, 1, 1, 1,
%e 15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 66, 1, 75, 1, 1, 1, 1,
%e 16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 99, 1, 28, 1, 1, 1, 1,
%e 17 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 75, 1, 42, 1, 1, 1, 1,
%e 18 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 63, 1, 1, 1, 1,
%e 19 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 48, 1, 1, 1, 1,
%e 20 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 63, 1, 71, 1, 1, 1, 1,
%o (PARI)
%o up_to = 105;
%o A000265(n) = (n>>valuation(n,2));
%o A371092(n) = floor((A000265(1+(3*n))+5)/6);
%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 A372287sq(n,k) = A371092(A372283sq(n,k));
%o A372287list(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] = A372287sq((a-(col-1)),col))); (v); };
%o v372287 = A372287list(up_to);
%o A372287(n) = v372287[n];
%Y Cf. A257852, A371092, A371094, A372282, A372283, A372288.
%Y Cf. also A371097 (array giving every fourth column, 1, 5, 9, ...), A371103 (array giving every even numbered column), also array A371101.
%K nonn,tabl
%O 1,10
%A _Antti Karttunen_, Apr 28 2024