A333242 Prime numbers with an odd number of steps in their prime index chain.

2, 5, 7, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 71, 73, 79, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 313, 317, 331, 337, 347, 349, 359, 373

Offset: 1

Author

Michael P. May, Mar 12 2020

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
• Michael P. May, Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence, arXiv:2112.08941 [math.GM], 2021.
• Michael P. May, Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.
• Michael P. May, Application of the Inclusion-Exclusion Principle to Prime Number Subsequences, arXiv:2402.13214 [math.GM], 2024.

Crossrefs

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

Programs

Maple

b:= proc(n) option remember;
       `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p; p:= a(n-1);
      do p:= nextprime(p);
         if b(p)::odd then break fi
      od; p
    end: a(1):=2:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 15 2020

Mathematica

Select[Prime@ Range@ 75, EvenQ@ Length@ NestWhileList[ PrimePi, #, PrimeQ] &] (* Giovanni Resta, Mar 15 2020 *)

PARI

\\ here b(n) is A078442.
b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k}
select(n->b(n)%2, [1..500]) \\ Andrew Howroyd, Mar 15 2020

Formula

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

a(n) = A000720(A262275(n)). - Andrew Howroyd, Mar 15 2020

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 15 2020