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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
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
Sequence in context: A027624 A165744 A330554 * A034900 A079388 A183605
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 March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)