A262275 Prime numbers with an even number of steps in their prime index chain.

3, 11, 17, 41, 67, 83, 109, 127, 157, 191, 211, 241, 277, 283, 353, 367, 401, 461, 509, 547, 563, 587, 617, 739, 773, 797, 859, 877, 967, 991, 1031, 1063, 1087, 1171, 1201, 1217, 1409, 1433, 1447, 1471, 1499, 1597, 1621, 1669, 1723, 1741, 1823, 1913, 2027, 2063, 2081, 2099, 2221, 2269, 2341, 2351

Offset

1,1

Comments

Old (incorrect) name was: Primes not appearing in A121543.

Number of terms less than 10^n: 1, 6, 30, 165, 1024, ... .

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1025 terms from Zak Seidov and Robert G. Wilson v)

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, Application of the Inclusion-Exclusion Principle to Prime Number Subsequences, arXiv:2402.13214 [math.GM], 2024.

Formula

From Alois P. Heinz, Mar 15 2020: (Start)

{ p in primes : A078442(p) mod 2 = 0 }.

a(n) = prime(A333242(n)). (End)

Example

11 is a term: 11 -> 5 -> 3 -> 2 -> 1, four (an even number of) steps "->" = pi = A000720.

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)::even then break fi
      od; p
    end: a(1):=3:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 15 2020

Mathematica

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]

Prog

(PARI) b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(prime, select(n->b(n)%2, [1..500])) \\ Michel Marcus, Jan 03 2022; after A333242

Crossrefs

Cf. A000040, A000720, A078442, A121543, A333242 (complement in primes).

Sequence in context: A240084 A100567 A270225 * A176804 A078116 A245045

Adjacent sequences: A262272 A262273 A262274 * A262276 A262277 A262278

Keyword

nonn

Author

Zak Seidov and Robert G. Wilson v, Sep 17 2015

Extensions

New name from Alois P. Heinz, Mar 15 2020

Status

approved