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 A073846 a(1) = 1; thereafter, every even-indexed term is prime and every odd-indexed term is composite. 17
 1, 2, 4, 3, 6, 5, 8, 7, 9, 11, 10, 13, 12, 17, 14, 19, 15, 23, 16, 29, 18, 31, 20, 37, 21, 41, 22, 43, 24, 47, 25, 53, 26, 59, 27, 61, 28, 67, 30, 71, 32, 73, 33, 79, 34, 83, 35, 89, 36, 97, 38, 101, 39, 103, 40, 107, 42, 109, 44, 113, 45, 127, 46, 131, 48, 137, 49, 139, 50 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equals A067747 shifted by one position. a(n) = A067747(n-1) [for n>1]. - R. J. Mathar, Apr 01 2007 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 Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 FORMULA a(2*n-1) = A018252(n); a(2*n) = A000040(n). - Reinhard Zumkeller, Jan 29 2014 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 (* 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 Cf. A002808, A018252, A000040, A261314. Sequence in context: A328047 A073898 A067747 * A110458 A217559 A217560 Adjacent sequences:  A073843 A073844 A073845 * A073847 A073848 A073849 KEYWORD easy,nonn AUTHOR Amarnath Murthy, Aug 14 2002 EXTENSIONS Edited by Robert G. Wilson v and Benoit Cloitre, Aug 16 2002 STATUS approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)