A073846 a(1) = 1; thereafter, every even-indexed term is prime and every odd-indexed term is composite.

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

Offset: 1

Author

Amarnath Murthy, Aug 14 2002

Keywords

• easy
• nonn

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
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A002808, A018252, A000040, A261314.

Programs

Haskell

import Data.List (transpose)
a073846 n = a073846_list !! (n-1)
a073846_list = concat $ transpose [a018252_list, a000040_list]
-- Reinhard Zumkeller, Jan 29 2014

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 *)

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

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

Extensions

Edited by Robert G. Wilson v and Benoit Cloitre, Aug 16 2002