A304524 - OEIS

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 (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) / (n3^A006667(n)) (see A127789).` `Conjecture: for all prime p, res(p) < res(993) = 2^61/(3^32993) = 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`