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%I #27 Jul 17 2018 16:24:48
%S 2,3,7,37,43,229,271,379,673,839,1987,5297,25111,44641,50221,94057,
%T 334423,1189057,1759579,2505337,28153249,46869157,87780541,584543567,
%U 768901097
%N Consider the ratio res(p) = 2^A006666(p) / (p*3^A006667(p)) where p is prime. The prime numbers in this sequence are those for which res(p) sets a new record.
%C Is the sequence finite?
%C In the general case, the residue of a number n in the 3x+1 problem is defined as the ratio res(n) = 2^A006666(n) / (n*3^A006667(n)) (see A127789).
%C Conjecture: for all prime p, res(p) < res(993) = 2^61/(3^32*993) = 1.253142... (see A304174).
%H Eric Roosendaal, <a href="http://www.ericr.nl/wondrous/index.html">On the 3x+1 Problem</a>
%e From _Jon E. Schoenfield_, May 23 2018: (Start)
%e Let D = A006666(p) and U = A006667(p); then res(p) = 2^D/(p*3^U). It seems clear that res(993) - res(p) is converging toward a positive value:
%e .
%e p | D | U | res(p) | res(993)-res(p)
%e ----------+-----+----+-----------------+----------------
%e 2 | 1 | 0 | 1 | 0.2531421443...
%e 3 | 5 | 2 | 1.1851851851... | 0.0679569592...
%e 7 | 11 | 5 | 1.2039976484... | 0.0491444959...
%e 37 | 15 | 6 | 1.2148444741... | 0.0382976702...
%e 43 | 20 | 9 | 1.2389111604... | 0.0142309838...
%e 229 | 24 | 10 | 1.2407145246... | 0.0124276197...
%e 271 | 29 | 13 | 1.2425797507... | 0.0105623936...
%e 379 | 39 | 19 | 1.2480350469... | 0.0051070974...
%e 673 | 43 | 21 | 1.2494773856... | 0.0036647587...
%e 839 | 56 | 29 | 1.2514151532... | 0.0017269911...
%e 1987 | 62 | 32 | 1.2525114739... | 0.0006306704...
%e 5297 | 65 | 33 | 1.2529055685... | 0.0002365758...
%e 25111 | 72 | 36 | 1.2529406796... | 0.0002014647...
%e 44641 | 76 | 38 | 1.2529625095... | 0.0001796348...
%e 50221 | 73 | 36 | 1.2529656281... | 0.0001765162...
%e 94057 | 85 | 43 | 1.2529812032... | 0.0001609411...
%e 334423 | 90 | 45 | 1.2529882803... | 0.0001538640...
%e 1189057 | 95 | 47 | 1.2529909733... | 0.0001511710...
%e 1759579 | 113 | 58 | 1.2529910420... | 0.0001511023...
%e 2505337 | 104 | 52 | 1.2529915763... | 0.0001505680...
%e 28153249 | 117 | 58 | 1.2529917096... | 0.0001504347...
%e 46869157 | 132 | 67 | 1.2529917720... | 0.0001503722...
%e 87780541 | 144 | 74 | 1.2529919281... | 0.0001502162...
%e 584543567 | 161 | 83 | 1.2529919325... | 0.0001502118...
%e 768901097 | 182 | 96 | 1.2529919396... | 0.0001502047...
%e (End)
%t lst={2};Print["a(n)"," ","A006667(a(n))"," ","A006666(a(n))"," ","res(a(n))"];q=1;Collatz[n_]:=NestWhileList[If[EvenQ[#],#/2,3 #+1]&,Prime[n],#>1&];nn=10000;t={};n=0;While[Length[t]<nn,n++;c=Collatz[n];ev=Length[Select[c,EvenQ]];od=Length[c]-ev-1;If[Prime[n]*3^od/2^ev<q,Print[Prime[n]," ",od," ",ev," ",N[2^ev/(Prime[n]*3^od),20]];AppendTo[lst,Prime[n]];If[n>5000,Break[]];q=Prime[n]*3^od/2^ev]];lst
%Y Cf. A006666, A006667, A127789, A304174.
%K nonn,more
%O 1,1
%A _Michel Lagneau_, May 14 2018
%E a(23)-a(24) from _Jon E. Schoenfield_, May 19 2018
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