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.

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

Author

Bernard Schott, Jun 09 2019

Keywords

• nonn

Comments

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

Examples

			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.
		

Crossrefs

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).

Programs

Mathematica

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

Formula

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