OFFSET
1,2
COMMENTS
From Chayim Lowen, Aug 12 2015: (Start)
Consider f(n,k) = a(f(n,k-1)) with f(n,0) = n. Let us also define f(n,k) for negative values of k as well using: f(n,k-1) = A073898(f(n,k)) where A073898(a(n)) = a(A073898(n)) = n. Let us denote the sequence {f(n,i)} for integers i by R#(n). It is clear that if a is a value in R#(b), R#(a) is just R#(b) with a different offset. Therefore, unless there is a need to do otherwise, let us denote each sequence by its lowest value. These sequences can only behave in one of two ways. They can either be periodic with f(n,m) = f(n,0) for some m, or they can include infinitely many distinct values. Here is the behavior of R#(n) for n<=100:
* R#(1), R#(2) and R#(9) are 1-cycles.
* R#(3), R#(5), R#(7), R#(10) and R#(12) are 2-cycles.
* R#(14), R#(62) and R#(84) are 3-cycles.
* R#(92) is a 6-cycle.
* R#(18) is a 22-cycle.
* R#(34) (A261314) has been checked up to f(34,86) = 1091595086717, R#(42) up to f(42,108) = 106838266736, R#(50) up to f(50,98) = 1078406742163, R#(60) up to f(60,80) = 765456394363, R#(74) up to f(74,78) = 687059343029, R#(82) up to f(82,75) = 682580868743 R#(86) up to f(86,74) = 182831963148, R#(88) up to f(88,66) = 719074799059, and R#(98) up to f(98,88) = 641383978721 without repeated values. Hence, their periods are either extremely large or nonexistent (infinite). I conjecture that the latter is the case. Note that these sequences are not necessarily all distinct as any two may simply be the same sequence with a large offset.
For all other n<=100, a(n) is included in one of the above sequences. (End)
Conjecturally, the integers that belong to one of these cycles are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 41, 43, 47, 53, 62, 84, 87, 92, 121, 127, 132, 135, 181, 199, 205, 317. - Michel Marcus, Mar 07 2021
LINKS
FORMULA
a(n) = A018252(ceiling(n/2))*A000035(n) + A000040(ceiling(n/2))*A059841(n), equivalent to Reinhard Zumkeller's formula. - Chayim Lowen, Jul 29 2015
a(2n)/a(2n-1) ~ log(n). - Thomas Ordowski, Sep 10 2015
MAPLE
N:= 100: # to get a(1) to a(2*N).
p:= ithprime(N):
P, NP:= selectremove(isprime, [$1..p]):
seq(op([NP[i], P[i]]), i=1..N); # Robert Israel, Dec 22 2014
MATHEMATICA
Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n]; Join[{1}, Flatten[ Transpose[{Table[Prime[n], {n, 1, 35}], Table[Composite[n], {n, 1, 35}]}]]]
f[upto_]:=Module[{prs=Prime[Range[PrimePi[upto]]], comps}, comps= Complement[ Range[upto], prs]; Riffle[Take[comps, Length[prs]], prs]]; f[150] (* Harvey P. Dale, Dec 03 2011 *)
PROG
(Haskell)
import Data.List (transpose)
a073846 n = a073846_list !! (n-1)
a073846_list = concat $ transpose [a018252_list, a000040_list]
-- Reinhard Zumkeller, Jan 29 2014
(PARI) c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = if (n==1, 1, if (n%2, c(n\2), prime(n/2))); \\ Michel Marcus, Mar 06 2021
(Python)
from sympy import prime, composite
def A073846(n): return 1 if n == 1 else (composite(n//2) if n % 2 else prime(n//2)) # Chai Wah Wu, Mar 09 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Amarnath Murthy, Aug 14 2002
EXTENSIONS
Edited by Robert G. Wilson v and Benoit Cloitre, Aug 16 2002
STATUS
approved