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
Keywords
Examples
11 is a term: 11 -> 5 -> 3 -> 2 -> 1, four (an even number of) steps "->" = pi = A000720.
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.
Programs
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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] -
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
Formula
From Alois P. Heinz, Mar 15 2020: (Start)
{ p in primes : A078442(p) mod 2 = 0 }.
a(n) = prime(A333242(n)). (End)
Extensions
New name from Alois P. Heinz, Mar 15 2020
Comments