cp's OEIS Frontend

A375006 Numbers whose Collatz trajectory is a Sidon sequence.

%I A375006 #36 Jul 31 2024 11:16:09
%S A375006 1,2,4,8,16,21,32,42,64,85,128,151,170,227,256,302,341,454,512,682,
%T A375006 1024,1365,2048,2730,4096,5461,8192,10922,14563,16384,21845,29126,
%U A375006 32768,43690,65536,87381,131072,174762,262144,349525,524288,699050,932067,1048576
%N A375006 Numbers whose Collatz trajectory is a Sidon sequence.
%C A375006 If a number is a term, then all its trajectory elements are terms. This can be used to accelerate the programs.
%C A375006 Each power of 2 is a term.
%C A375006 Each term that is a multiple of 4 is a power of 2: The trajectory of 2^r(2s+1) with r > 1 is 2^r(2s+1), ..., 2s+1, 6s+4, ..., 1, which includes 4(2s+1), and 4(2s+1) + 1 = 2s+1 + 6s+4, so 2^r(2s+1) cannot be a term unless s = 0.
%C A375006 Each even term is the double of an odd term or a power of 2: If k is an even term, then k/2 is a term. If k/2 is odd, then k is the double of an odd term. If k/2 is even, then k is a multiple of 4 and hence a power of 2.
%C A375006 Hugo Pfoertner conjectured that the double of each odd term is a term, too. If this is true, then A375006 is the union of A374527, twice A374527, and A000079.
%C A375006 Level-wise traversal of the corresponding subtree of the Collatz tree allows one to quickly generate many (unsorted) terms of the sequence. Up to at least level 1000 there is no counterexample to the conjectured occurrence of twice of all terms of A374527 in the present sequence. The results also suggest that A374527 is not finite.
%H A375006 Markus Sigg, <a href="/A375006/b375006.txt">Table of n, a(n) for n = 1..77</a>
%H A375006 OEIS Wiki, <a href="https://oeis.org/wiki/3x%2B1_problem">3x+1 problem, Collatz trajectories</a>.
%H A375006 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sidon_sequence">Sidon sequence</a>.
%e A375006 3 is not a term because its trajectory is {3,10,5,16,8,4,2,1} and 3+10 = 5+8.
%o A375006 (PARI) is_A375006(k) = { my(T=List([k]), S=Set([2*k])); while(k>1, k=if(k%2==0,k/2,3*k+1); listput(T,k); for(i=1,#T, my(s=T[i]+k); if(setsearch(S,s),return(0),S=setunion(S,Set([s])));); ); 1 };
%o A375006 (Python)
%o A375006 from itertools import count, islice
%o A375006 def A375006_gen(startvalue=1): # generator of terms >= startvalue
%o A375006     for n in count(max(startvalue,1)):
%o A375006         t, a, c = [n], n, set()
%o A375006         while a > 1:
%o A375006             a = 3*a+1 if a&1 else a>>1
%o A375006             for p in t:
%o A375006                 if (b:=p+a) in c:
%o A375006                     break
%o A375006                 c.add(b)
%o A375006             else:
%o A375006                 t.append(a)
%o A375006                 continue
%o A375006             break
%o A375006         else:
%o A375006             yield n
%o A375006 A375006_list = list(islice(A375006_gen(startvalue=1),20)) # _Chai Wah Wu_, Jul 27 2024
%Y A375006 Cf. A000079, A374527.
%K A375006 nonn
%O A375006 1,2
%A A375006 _Markus Sigg_, Jul 27 2024