A380757 Powers of primes that have a primitive root.

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset

1,2

Comments

Proper subset of A033948.

A046022 is a proper subset of this sequence.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Union of {1, 2, 4} and A061345.

This sequence is A000961 without A000079(k) for k > 2.

A033948 = union of {a(n)} and {2*a(n)} without 8 = union of {a(n)} and A278568, where {a(n)} represents this sequence.

Intersection of A000961 and A033948.

Define c(m) to be the number of terms that do not exceed m. Then for m >= 4, c(m) = 3 + (Sum_{k = 1..floor(log_2(m))} pi(floor(m^(1/k)))) - floor(log_2(m)) = 3 + A065515(m) - A113473(m).

Mathematica

With[{nn = 2^8},
  Complement[#, Array[2^# &, Floor@ Log2[#[[-1]]] + 2, 3]] &@
  Union[{1}, Prime@ Range@ PrimePi[#[[-1]] ], #] &@
  Select[Union@ Flatten@
    Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[nn/b^3]}],
    PrimePowerQ] ]

Prog

(Python)
from sympy import primepi, integer_nthroot
def A380757(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n if x<6 else int(n+x-3-sum(primepi(integer_nthroot(x, k)[0])-1 for k in range(1, x.bit_length())))
    return bisection(f, n, n) # Chai Wah Wu, Feb 03 2025

Crossrefs

Cf. A000079, A000961, A033948, A046022, A061345, A278568.

Sequence in context: A325266 A284288 A343983 * A074583 A001092 A327782

Adjacent sequences: A380754 A380755 A380756 * A380758 A380759 A380760

Keyword

nonn,easy

Author

Michael De Vlieger, Feb 01 2025

Status

approved