A067747 -id:A067747 - cp's OEIS frontend

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

Amarnath Murthy, Aug 14 2002

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A002808, A018252, A000040, A261314.

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

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

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 *)

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

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

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

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

A308598 The smaller term of the pair (a(n), a(n+1)) is always prime and in each pair there is a composite number; a(1) = 2 and the sequence is always extended with the smallest integer not yet present and not leading to a contradiction.

Original entry on oeis.org

2, 4, 3, 6, 5, 8, 7, 12, 11, 14, 13, 18, 17, 20, 19, 24, 23, 30, 29, 32, 31, 38, 37, 42, 41, 44, 43, 48, 47, 54, 53, 60, 59, 62, 61, 68, 67, 72, 71, 74, 73, 80, 79, 84, 83, 90, 89, 98, 97, 102, 101, 104, 103, 108, 107, 110, 109, 114, 113, 128, 127, 132, 131, 138, 137, 140, 139, 150, 149

Offset: 1

Bernard Schott, Jun 09 2019

nonn

The idea of this sequence comes from A282649 where "larger" replaces "smaller".
The sequence is not a permutation of the positive integers.
The 1st bisection is A000040 (the primes) and the 2nd bisection is A008864 \ {3} (prime(n) + 1).
Consecutive primes p < q separated by composites c = q + 1. - Michael De Vlieger, Jun 09 2019

			In the 1st pair of integers (2,4) the smaller term is (2), which is prime;
In the 2nd pair of integers (4,3) the smaller term is (3), which is prime;
In the 3rd pair of integers (3,6) the smaller term is (3), which is prime;
In the 4th pair of integers (6,5) the smaller term is (5), which is prime;
In the 5th pair of integers (5,8) the smaller term is (5), which is prime; etc.

Cf. A000040 (prime numbers), A002808 (composite numbers), A008864 (prime(n) + 1).
Cf. A282649 (similar, with larger term).
Cf. A067747, A073846, A073898 (sequences with same start).

Fold[Join[#1, {#2, NextPrime@ #2 + 1}] &, {#, NextPrime@ # + 1} &@ 2, Prime@ Range[2, 35]] (* Michael De Vlieger, Jun 09 2019 *)

n odd: a(n) = prime((n+1)/2) = A000040((n+1)/2).
n even: a(n) = a(n+1) + 1 = prime(n/2 + 1) + 1 = A008864(n/2 + 1).
Alternatively, if a(n-1) is prime, a(n) = 1 + min prime > a(n-1) else a(n) = a(n-1) - 1. - Bill McEachen, May 16 2024

cp's OEIS Frontend

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

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