A376198 a(1) = 1, a(2) = 2. Thereafter, let smc and smp denote the smallest missing composite and smallest missing prime. If a(n) is composite, then if a(n) = 2*smp then a(n+1) = smp, otherwise a(n+1) = smc; if a(n) is a prime, then if smp < smc, a(n+1) = smp, otherwise a(n+1) = smc.

1, 2, 3, 4, 6, 8, 9, 10, 5, 7, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 13, 17, 19, 23, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 29, 31, 37, 41, 43, 47, 53, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94

Offset

1,2

Comments

The composite terms appear in their natural order, as do the primes.

This is a simplified version of A375564 (the difference being in the way the composite numbers are handled: here they appear in order, whereas in A375564 successive composite numbers must have a common gcd greater than 1).

The following table was calculated by Michael S. Branicky on Oct 04 2024.

It shows the beginning, end, and length of the k-th run of successive primes.

a b c : d e f [a = k, b = A376750(k), c = A376751(k),

1 2 2 : 3 3 2 d = A376752(k), e = A376753(k), f = A376754(k)]

2 9 5 : 11 11 3

3 23 13 : 26 23 4

4 52 29 : 59 59 8

5 110 61 : 122 113 13

6 231 127 : 254 251 24

7 472 257 : 514 509 43

8 965 521 : 1042 1039 78

9 1958 1049 : 2099 2099 142

10 3962 2111 : 4222 4219 261

11 7980 4229 : 8458 8447 479

12 16029 8461 : 16922 16921 894

13 32181 16927 : 33854 33851 1674

14 64597 33857 : 67714 67709 3118

15 129574 67723 : 135446 135433 5873

16 259798 135449 : 270899 270899 11102

17 520835 270913 : 541826 541817 20992

18 1043833 541831 : 1083662 1083659 39830

19 2091473 1083689 : 2167378 2167369 75906

20 4190135 2167393 : 4334786 4334777 144652

21 8392863 4334791 : 8669582 8669543 276720

22 16809322 8669593 : 17339186 17339177 529865

23 33661860 17339197 : 34678394 34678381 1016535

24 67402676 34678421 : 69356842 69356839 1954167

25 134952624 69356857 : 138713714 138713711 3761091

26 270177158 138713717 : 277427434 277427431 7250277

27 540861852 277427441 : 554854882 554854873 13993031

28 1082667610 554854889 : 1109709778 1109709709 27042169

29 2167106199 1109709791 : 2219419582 2219419577 52313384

30 4337519113 2219419597 : 4438839194 4438839173 101320082

31 8681255531 4438839259 : 8877678518 8877678499 196422988

32 17374202846 8877678527 : 17755357054 17755357051 381154209

33 34770433922 17755357069 : 35510714138 35510714137 740280217

Links

Michael S. Branicky, Table of n, a(n) for n = 1..100000

Prog

(Python)
from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    an, smc, smp = 2, 4, 3
    yield from [1, 2]
    while True:
        if not isprime(an):
            an = smp if an == 2*smp else smc
        else:
            an = smp if smp < smc else smc
        if an == smp: smp = nextprime(smp)
        else:
            smc += 1
            while isprime(smc): smc += 1
        yield an
print(list(islice(agen(), 87))) # Michael S. Branicky, Oct 03 2024

Crossrefs

Cf. A375564, A376199-A376201, A376750-A376754.

See also A113646 (next composite number).

Sequence in context: A067118 A320503 A138561 * A333841 A231236 A346131

Adjacent sequences: A376195 A376196 A376197 * A376199 A376200 A376201

Keyword

nonn

Author

N. J. A. Sloane, Oct 03 2024

Status

approved