A337144 -id:A337144 - cp's OEIS frontend

A380244 The Collatz (or 3x+1) trajectory starting at a(n) contains exactly n odd integers and a(n) is the n-th number with this property.

1, 10, 12, 68, 45, 30, 72, 101, 134, 179, 237, 314, 422, 551, 723, 509, 1282, 887, 1170, 1535, 2021, 1509, 1899, 2412, 1780, 2217, 3170, 3867, 2819, 3728, 2511, 3155, 3972, 2802, 3578, 2623, 3444, 4302, 3087, 3968, 2690, 1806, 2336, 1593, 2084, 2757, 1884, 2477

Offset: 1

Alois P. Heinz, Jan 17 2025

nonn

			a(2) = 10 is the second integer (after 5) having exactly two odd integers in the Collatz trajectory: 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.

Alois P. Heinz, Table of n, a(n) for n = 1..235
Wikipedia, Collatz Conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Main diagonal of A354236.

Cf. A006577, A006667, A078719, A337144.

b:= proc(n) option remember; irem(n, 2, 'r')+
      `if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
    end:
A:= proc() local h, p, q; p, q:= proc() [] end, 0;
      proc(n, k)
        if k=1 then return 2^(n-1) fi;
        while nops(p(k)) A(n$2):
seq(a(n), n=1..48);

A078719(a(n)) = n.

A337149 Positive integers k such that the number of steps it takes to reach 1 in the '3x+1' problem is different for all j < k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 22, 24, 25, 27, 28, 31, 33, 34, 36, 39, 41, 43, 47, 48, 49, 54, 57, 62, 65, 71, 72, 73, 78, 82, 86, 91, 94, 97, 98, 103, 105, 107, 108, 111, 114, 121, 123, 124, 129, 130, 135, 137, 142, 145, 153, 155, 159

Offset: 1

Alois P. Heinz, Jan 27 2021

Positive integers k such that A337144(k) = 1.
Or positive integers k such that A006577(k) != A006577(j) for all j = 1..k-1.
Different from A129304.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Wikipedia, Collatz Conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577, A129304, A337144, A337150.

collatz:= proc(n) option remember; `if`(n=1, 0,
   1 + collatz(`if`(n::even, n/2, 3*n+1)))
end:
b:= proc() 0 end:
g:= proc(n) option remember; local t;
     `if`(n=1, 0, g(n-1));
      t:= collatz(n); b(t):= b(t)+1
    end:
a:= proc(n) option remember; local k; for k
      from 1+a(n-1) while g(k)>1 do od; k
    end: a(0):=0:
seq(a(n), n=1..100);

collatz[n_] := collatz[n] = If[n==1, 0,
   1+collatz[If[EvenQ[n], n/2, 3n+1]]];
b[_] = 0;
g[n_] := g[n] = Module[{t}, If[n==1, 0, g[n-1]];
   t = collatz[n]; b[t] = b[t]+1];
a[n_] := a[n] = Module[{k}, For[k = 1+a[n-1],
   g[k] > 1, k++]; k]; a[0] = 0;
Array[a, 100] (* Jean-François Alcover, Jan 30 2021, after Alois P. Heinz *)

A006577(a(n)) = A337150(n).

cp's OEIS Frontend

A380244 The Collatz (or 3x+1) trajectory starting at a(n) contains exactly n odd integers and a(n) is the n-th number with this property.

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