A304524 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.

2, 3, 7, 37, 43, 229, 271, 379, 673, 839, 1987, 5297, 25111, 44641, 50221, 94057, 334423, 1189057, 1759579, 2505337, 28153249, 46869157, 87780541, 584543567, 768901097

Offset: 1

Author

Michel Lagneau, May 14 2018

Keywords

• nonn
• more

Comments

Is the sequence finite?

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).

Conjecture: for all prime p, res(p) < res(993) = 2^61/(3^32*993) = 1.253142... (see A304174).

Examples

			From _Jon E. Schoenfield_, May 23 2018: (Start)
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:
.
          p |   D |  U |     res(p)      | res(993)-res(p)
  ----------+-----+----+-----------------+----------------
          2 |   1 |  0 | 1               | 0.2531421443...
          3 |   5 |  2 | 1.1851851851... | 0.0679569592...
          7 |  11 |  5 | 1.2039976484... | 0.0491444959...
         37 |  15 |  6 | 1.2148444741... | 0.0382976702...
         43 |  20 |  9 | 1.2389111604... | 0.0142309838...
        229 |  24 | 10 | 1.2407145246... | 0.0124276197...
        271 |  29 | 13 | 1.2425797507... | 0.0105623936...
        379 |  39 | 19 | 1.2480350469... | 0.0051070974...
        673 |  43 | 21 | 1.2494773856... | 0.0036647587...
        839 |  56 | 29 | 1.2514151532... | 0.0017269911...
       1987 |  62 | 32 | 1.2525114739... | 0.0006306704...
       5297 |  65 | 33 | 1.2529055685... | 0.0002365758...
      25111 |  72 | 36 | 1.2529406796... | 0.0002014647...
      44641 |  76 | 38 | 1.2529625095... | 0.0001796348...
      50221 |  73 | 36 | 1.2529656281... | 0.0001765162...
      94057 |  85 | 43 | 1.2529812032... | 0.0001609411...
     334423 |  90 | 45 | 1.2529882803... | 0.0001538640...
    1189057 |  95 | 47 | 1.2529909733... | 0.0001511710...
    1759579 | 113 | 58 | 1.2529910420... | 0.0001511023...
    2505337 | 104 | 52 | 1.2529915763... | 0.0001505680...
   28153249 | 117 | 58 | 1.2529917096... | 0.0001504347...
   46869157 | 132 | 67 | 1.2529917720... | 0.0001503722...
   87780541 | 144 | 74 | 1.2529919281... | 0.0001502162...
  584543567 | 161 | 83 | 1.2529919325... | 0.0001502118...
  768901097 | 182 | 96 | 1.2529919396... | 0.0001502047...
(End)
		

Links

• Eric Roosendaal, On the 3x+1 Problem

Crossrefs

Cf. A006666, A006667, A127789, A304174.

Programs

Mathematica

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]5000,Break[]];q=Prime[n]*3^od/2^ev]];lst

Extensions

a(23)-a(24) from Jon E. Schoenfield, May 19 2018