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%I #18 Oct 20 2021 21:30:30
%S 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,
%T 57,41,58,41,59,78,59,79,100,79,101,124,101,125,101,126,101,127,154,
%U 127,155,127,156,127,157,188,157,189,157,190,157,191,226,191,227,264,227,265
%N 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.
%C Without repeated terms, the primes appear in order as A070865.
%C 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).
%C For n >= 3: When a(n) = prime p it is followed by an even number j at a(n+1); p repeats k-j times (where k is the smallest prime > j), appearing at a(n+2m) {m=1..k-j}. 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.
%H Alois P. Heinz, <a href="/A284188/b284188.txt">Table of n, a(n) for n = 1..100000</a>
%H Michael De Vlieger, <a href="/A284188/a284188.png">Log-log scatterplot of a(n)</a> for n=1..2^16.
%H Michael De Vlieger, <a href="/A284188/a284188_1.png">Log-log scatterplot of a(n)</a> 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.
%e 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) = 18-7 = 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.
%p c:= proc(n) option remember; `if`(n<1, 0,
%p `if`(isprime(a(n)), 1, 0)+c(n-1))
%p end:
%p a:= proc(n) option remember; `if`(n=1, 2, (m->
%p `if`(isprime(m), 1, -1)*c(n-1)+m)(a(n-1)))
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Apr 15 2017
%t 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 *)
%o (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
%Y Cf. A070865, A284172.
%K nonn
%O 1,1
%A _Bob Selcoe_, Mar 21 2017
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