

A284188


a(1)=2; thereafter a(n+1) = a(n)+i if a(n) is a prime and a(1),...,a(n) contains i primes, or a(n+1) = a(n)i if a(n) is composite and a(1),...,a(n) contains i primes.


2



2, 3, 5, 8, 5, 9, 5, 10, 5, 11, 18, 11, 19, 28, 19, 29, 40, 29, 41, 54, 41, 55, 41, 56, 41, 57, 41, 58, 41, 59, 78, 59, 79, 100, 79, 101, 124, 101, 125, 101, 126, 101, 127, 154, 127, 155, 127, 156, 127, 157, 188, 157, 189, 157, 190, 157, 191, 226, 191, 227, 264, 227, 265
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OFFSET

1,1


COMMENTS

Without repeated terms, the primes appear in order as A070865.
Variant of A284172; the difference is that in A284172, a(n+1) = a(n)i if a(n) is composite and a(1),...,a(n) contains i composites (rather than i primes).
For n >= 3: When a(n) = prime p it is followed by an even number j at a(n+1); p repeats kj times (where k is the smallest prime > j), appearing at a(n+2m) {m=1..kj}. a(n+2m+1) = p+m until p+m = k (immediately following the final p); k now becomes "new p" immediately followed by a "new j", and the process repeats.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Michael De Vlieger, Loglog scatterplot of a(n) for n=1..2^16.
Michael De Vlieger, Loglog scatterplot of a(n) for n=1..2^12, showing a(n) in red and A284172(n) in blue, accentuating the first 12 terms of A284172(n) by enlargement.


EXAMPLE

a(10) = 11; there are 7 primes in the sequence up to and including a(10) so a(11) = 11+7 = 18. 18 is composite so a(12) = 187 = 11. Now there are 8 primes in the sequence; and since 11 is prime, a(13) = 11+8 = 19 (the 9th prime in the sequence), so a(14) = 28.


MAPLE

c:= proc(n) option remember; `if`(n<1, 0,
`if`(isprime(a(n)), 1, 0)+c(n1))
end:
a:= proc(n) option remember; `if`(n=1, 2, (m>
`if`(isprime(m), 1, 1)*c(n1)+m)(a(n1)))
end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 15 2017


MATHEMATICA

Block[{c = 1, m = 2, n}, {2}~Join~Reap[Do[If[PrimeQ[m], Set[n, m + c]; c++, Set[n, m  c + 1]]; Sow[n]; m = n, 63]][[1, 1]]] (* Michael De Vlieger, Oct 20 2021 *)


PROG

(PARI) lista(nn) = {print1(a=2, ", "); nbp = 1; for (n=2, nn, if (isprime(a), a += nbp, a = nbp); print1(a, ", "); if (isprime(a), nbp++); ); } \\ Michel Marcus, Mar 24 2017


CROSSREFS

Cf. A070865, A284172.
Sequence in context: A191450 A257122 A249188 * A011158 A104647 A065115
Adjacent sequences: A284185 A284186 A284187 * A284189 A284190 A284191


KEYWORD

nonn


AUTHOR

Bob Selcoe, Mar 21 2017


STATUS

approved



