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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

LINKS

Table of n, a(n) for n=1..25.

Eric Roosendaal, On the 3x+1 Problem

EXAMPLE

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)

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]<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

CROSSREFS

Cf. A006666, A006667, A127789, A304174.

Sequence in context: A027624 A165744 A330554 * A034900 A079388 A183605

Adjacent sequences:  A304521 A304522 A304523 * A304525 A304526 A304527

KEYWORD

nonn,more

AUTHOR

Michel Lagneau, May 14 2018

EXTENSIONS

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

STATUS

approved

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Last modified February 21 18:05 EST 2020. Contains 332107 sequences. (Running on oeis4.)