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%I #45 Feb 21 2024 01:11:10
%S 3,11,17,41,67,83,109,127,157,191,211,241,277,283,353,367,401,461,509,
%T 547,563,587,617,739,773,797,859,877,967,991,1031,1063,1087,1171,1201,
%U 1217,1409,1433,1447,1471,1499,1597,1621,1669,1723,1741,1823,1913,2027,2063,2081,2099,2221,2269,2341,2351
%N Prime numbers with an even number of steps in their prime index chain.
%C Old (incorrect) name was: Primes not appearing in A121543.
%C Number of terms less than 10^n: 1, 6, 30, 165, 1024, ... .
%H Alois P. Heinz, <a href="/A262275/b262275.txt">Table of n, a(n) for n = 1..10000</a> (first 1025 terms from Zak Seidov and Robert G. Wilson v)
%H Michael P. May, <a href="https://doi.org/10.35834/2020/3202158">Properties of Higher-Order Prime Number Sequences</a>, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and <a href="https://arxiv.org/abs/2108.04662">arXiv version</a>, arXiv:2108.04662 [math.NT], 2021.
%H Michael P. May, <a href="https://arxiv.org/abs/2402.13214">Application of the Inclusion-Exclusion Principle to Prime Number Subsequences</a>, arXiv:2402.13214 [math.GM], 2024.
%F From _Alois P. Heinz_, Mar 15 2020: (Start)
%F { p in primes : A078442(p) mod 2 = 0 }.
%F a(n) = prime(A333242(n)). (End)
%e 11 is a term: 11 -> 5 -> 3 -> 2 -> 1, four (an even number of) steps "->" = pi = A000720.
%p b:= proc(n) option remember;
%p `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
%p end:
%p a:= proc(n) option remember; local p; p:= a(n-1);
%p do p:= nextprime(p);
%p if b(p)::even then break fi
%p od; p
%p end: a(1):=3:
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Mar 15 2020
%t fQ[n_] := If[ !PrimeQ[n] || (PrimeQ[n] && FreeQ[lst, PrimePi[n]]), AppendTo[lst, n]]; k = 2; lst = {1}; While[k < 2401, fQ@ k; k++]; Select[lst, PrimeQ]
%o (PARI) b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
%o apply(prime, select(n->b(n)%2, [1..500])) \\ _Michel Marcus_, Jan 03 2022; after A333242
%Y Cf. A000040, A000720, A078442, A121543, A333242 (complement in primes).
%K nonn
%O 1,1
%A _Zak Seidov_ and _Robert G. Wilson v_, Sep 17 2015
%E New name from _Alois P. Heinz_, Mar 15 2020
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