A352374 a(1) = 1. For n >= 2, a(n) is the least nonprime not already in the sequence such that there is at least one prime between a(n-1) and a(n).

1, 4, 6, 8, 12, 9, 14, 10, 15, 18, 16, 20, 24, 21, 25, 22, 26, 30, 27, 32, 28, 33, 38, 34, 39, 35, 40, 36, 42, 44, 48, 45, 49, 46, 50, 54, 51, 55, 52, 56, 60, 57, 62, 58, 63, 68, 64, 69, 65, 70, 66, 72, 74, 80, 75, 81, 76, 82, 77, 84, 78, 85, 90, 86, 91, 87, 92, 88, 93, 98

Offset

1,2

Comments

Inspired by Recamán's sequence A005132.

This is a permutation of the nonprime numbers.

Is the number of primes that may fall between any two consecutive terms bounded?

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

Following a(4) = 8, a(5) cannot be 9 or 10 since no prime number falls between 8 and these. Therefore, a(5) = 12, as it is nonprime and prime 11 falls between it and 8.
Then, a(6) = 9 (even though 9 is less than 12), as prime 11 falls between it and 12.

Mathematica

nn = 120; c[_] = 0; a[1] = c[1] = u = 1; m = 0; While[c[u] > 0, u++]; Do[k = u; While[Set[p, PrimePi[k]]; Nand[c[k] == 0, CompositeQ[k], Abs[p - m] > 0], k++]; If[k == u, While[c[u] > 0, u++]]; Set[{a[i], c[k], m}, {k, i, p}], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Apr 16 2022 *)

Prog

(MATLAB)
function [ a ] = A352374( max_n )
     a(1) = 1;
     for n=2:max_n
         k = 2;
         while isprime(k) ...
             ||~isempty(find(a==k, 1)) ...
             ||isempty(find(isprime(min(k, a(end)):max(k, a(end))), 1))
             k = k+1;
         end
         a(n) = k;
     end
end % Thomas Scheuerle, Mar 14 2022
(Python)
from itertools import islice
from sympy import isprime, nextprime
def primebetween(k, m): return nextprime(min(k, m)) < max(k, m)
def agen(): # generator of terms
    lastan, an, aset, leastout = None, 1, set(), 2
    while True:
        yield an
        aset.add(an)
        lastan, an = an, leastout
        while an in aset or isprime(an) or not primebetween(lastan, an):
            an += 1
        while leastout in aset: leastout += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Mar 14 2022

Crossrefs

Cf. A005132.

Sequence in context: A320130 A320128 A320127 * A110646 A320126 A234520

Adjacent sequences: A352371 A352372 A352373 * A352375 A352376 A352377

Keyword

nonn

Author

Tamas Sandor Nagy, Mar 14 2022

Status

approved