A333242 - OEIS

%I #42 Feb 21 2024 01:10:57

%S 2,5,7,13,19,23,29,31,37,43,47,53,59,61,71,73,79,89,97,101,103,107,

%T 113,131,137,139,149,151,163,167,173,179,181,193,197,199,223,227,229,

%U 233,239,251,257,263,269,271,281,293,307,311,313,317,331,337,347,349,359,373

%N Prime numbers with an odd number of steps in their prime index chain.

%C This sequence can also be generated by the N-sieve.

%H Alois P. Heinz, <a href="/A333242/b333242.txt">Table of n, a(n) for n = 1..10000</a>

%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/2112.08941">Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence</a>, arXiv:2112.08941 [math.GM], 2021.

%H Michael P. May, <a href="https://doi.org/10.35834/2023/3501105">Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence</a>, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.

%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 { p in primes : A078442(p) mod 2 = 1 }.

%F a(n) = A000720(A262275(n)). - _Andrew Howroyd_, Mar 15 2020

%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)::odd then break fi

%p       od; p

%p     end: a(1):=2:

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 15 2020

%t Select[Prime@ Range@ 75, EvenQ@ Length@ NestWhileList[ PrimePi, #, PrimeQ] &] (* _Giovanni Resta_, Mar 15 2020 *)

%o (PARI) \\ here b(n) is A078442.

%o b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k}

%o select(n->b(n)%2, [1..500]) \\ _Andrew Howroyd_, Mar 15 2020

%Y Cf. A000040, A000720, A078442, A262275 (complement in primes), A333243, A333244.

%K nonn

%O 1,1

%A _Michael P. May_, Mar 12 2020

%E Terms a(21) and beyond from _Andrew Howroyd_, Mar 15 2020