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

1,1

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.

Formula

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A169690 A144300 A258430 * A045353 A038985 A109652

Adjacent sequences: A333239 A333240 A333241 * A333243 A333244 A333245

Keyword

nonn

Author

Michael P. May, Mar 12 2020

Extensions

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

Status

approved