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%I A372288 #15 May 10 2024 12:29:23
%S A372288 1,1,1,1,1,1,1,1,1,3,1,1,1,3,3,1,1,1,3,3,1,1,1,1,3,3,3,3,1,1,1,1,3,3,
%T A372288 1,3,1,1,1,1,3,1,1,3,3,1,1,1,1,1,1,1,5,3,3,1,1,1,1,1,1,1,3,1,3,1,1,1,
%U A372288 1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1,3,1,5,3,1,1,1,1,1,1,1,1,1,3,1,3,5,3
%N A372288 Array read by upward antidiagonals: A(n, k) = A265745(A372282(n, k)), n,k >= 1, where A265745(n) is the sum of digits in "Jacobsthal greedy base".
%C A372288 Collatz conjecture is equal to the claim that each column will eventually settle to constant 1's, somewhere under its topmost row. This works as only the bisection A002450 of Jacobsthal numbers (A001045) contains numbers of the form 4k+1, while the other bisection contains only numbers of the form 4k+3, which do not occur among the range of A372351. See also the comments in A371094.
%H A372288 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A372288 Array begins:
%e A372288 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
%e A372288 ---+----------------------------------------------------------------------------
%e A372288 1 | 1, 1, 1, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 1,
%e A372288 2 | 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 5, 5, 5, 3, 5, 3, 3, 3, 5, 5, 3,
%e A372288 3 | 1, 1, 1, 3, 3, 3, 1, 5, 1, 3, 1, 3, 3, 5, 3, 5, 5, 1, 3, 3, 5, 3,
%e A372288 4 | 1, 1, 1, 3, 3, 1, 1, 3, 1, 3, 1, 1, 3, 5, 3, 3, 3, 1, 3, 5, 5, 3,
%e A372288 5 | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 5, 3, 1, 3, 3, 3, 3,
%e A372288 6 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 5, 3, 1, 1, 5, 5, 3,
%e A372288 7 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 3, 1, 1, 3, 5, 3,
%e A372288 8 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 3, 3, 3,
%e A372288 9 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 5, 1,
%e A372288 10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 3, 5, 1,
%e A372288 11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2155, 1, 1, 1, 1, 5, 1,
%e A372288 12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 6251, 1,
%e A372288 13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10347, 1, 5, 1, 1, 1, 1, 5, 1,
%e A372288 14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 5, 1,
%e A372288 15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 5, 1,
%e A372288 16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 7, 1,
%o A372288 (PARI)
%o A372288 up_to = 105;
%o A372288 A130249(n) = (#binary(3*n+1)-1);
%o A372288 A001045(n) = (2^n - (-1)^n) / 3;
%o A372288 A265745(n) = { my(s=0); while(n,s++; n -= A001045(A130249(n))); (s); };
%o A372288 A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
%o A372288 A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
%o A372288 A372288sq(n,k) = A265745(A372282sq(n,k));
%o A372288 A372288list(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] = A372288sq((a-(col-1)),col))); (v); };
%o A372288 v372288 = A372288list(up_to);
%o A372288 A372288(n) = v372288[n];
%Y A372288 Cf. A001045, A002450, A265745, A265747, A371094, A372351, A372282, A372283, A372287.
%Y A372288 Cf. also array A372561 (formed by columns whose indices in this array are given by A372443).
%K A372288 nonn,tabl
%O A372288 1,10
%A A372288 _Antti Karttunen_, Apr 28 2024