A387335 1, 2, 4, along with numbers divisible by at least two distinct primes.

1, 2, 4, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset

1,2

Comments

The Jordan totient function is defined by the formula J_k(n) = n^k * Product{p prime divisor of n} (1-1/p^k). Because of the combinatorial origin of this function, it was originally understood that k must be a natural number. However, the formula still provides usable values for all positive real values of k.

Now define the Jordan cototient function by CoJ_k(n) = n^k - J_k(n). Then it appears that the terms of this sequence are precisely the values of n for which CoJ_k(n) achieves a new record for sufficiently small positive k. This includes 1, 2, 4, and the numbers that are not prime powers (A024619).

Links

James C. McMahon, Table of n, a(n) for n = 1..10000

Mathematica

Union[{2, 4}, Select[Range[110], Not@*PrimePowerQ]] (* James C. McMahon, Dec 05 2025 *)

Prog

(Python)
from sympy import primepi, integer_nthroot
def A387335(n):
    def f(x): return int(n+1-min(3, y:=x.bit_length())+sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, y)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 05 2025

Crossrefs

Cf. A024619, A361102.

Sequence in context: A090127 A388841 A057910 * A336066 A120351 A022305

Adjacent sequences: A387332 A387333 A387334 * A387336 A387337 A387338

Keyword

nonn

Author

Hal M. Switkay, Nov 30 2025

Status

approved