A393213 Primes and cubes of primes.

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset

1,1

Comments

Subsequence of A186285 and first differs from it at n = 102: A186285(102) = 512 = 2^9 is not a term of this sequence.

Numbers with an equal number of exponential and non-exponential divisors: numbers k such that A049419(k) = A160097(k).

The numbers with more exponential divisors than non-exponential divisors are 1 and the squares of the terms in this sequence, i.e., 1, squares of primes (A001248), and fourth powers of primes (A030514).

{2, 3} and {7, 8} are the only pairs of consecutive integers that are both terms in this sequence.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

A prime p has 2 divisors, 1 and p. p is an exponential divisor, and 1 is a non-exponential divisor.
A cube of a prime p^3 has 4 divisors, 1, p, p^2 and p^3. p and p^3 are exponential divisors, and 1 and p^2 are non-exponential divisors.

Mathematica

With[{m = 60}, Union[Prime[Range[m]], Select[Range[Surd[Prime[m], 3]], PrimeQ]^3]]

Prog

(PARI) list(lim) = Set(concat(primes(primepi(lim)), apply(x -> x^3, primes(primepi(sqrtnint(lim, 3))))));
(Python)
from sympy import primepi, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A393213(n):
    def f(x): return n+x-primepi(x)-primepi(integer_nthroot(x, 3)[0])
    return bisection(f, n, n) # Chai Wah Wu, Feb 06 2026

Crossrefs

Disjoint union of A000040 and A030078.

Subsequence of A186285.

Cf. A049419, A001248, A030514, A160097, A274205, A322791.

Similar sequences: A016825, A048109, A348097, A356413.

Sequence in context: A174895 A375270 A186285 * A378170 A375886 A190855

Adjacent sequences: A393210 A393211 A393212 * A393214 A393215 A393216

Keyword

nonn,easy

Author

Amiram Eldar, Feb 06 2026

Status

approved