A059404 Numbers with different exponents in their prime factorizations.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189

Offset

1,1

Comments

Former name: Numbers k such that k/(largest power of squarefree kernel of k) is larger than 1.

Complement of A072774 (powers of squarefree numbers).

Also numbers k = p(1)^alpha(1)* ... * p(r)^alpha(r) such that RootMeanSquare(alpha(1), ..., alpha(r)) is not an integer. - Ctibor O. Zizka, Sep 19 2008

Links

Donald Alan Morrison, Table of n, a(n) for n = 1..10000

Donald Alan Morrison, Sage program

Formula

A062760(a(n)) > 1, i.e., a(n)/(A007947(a(n))^A051904(a(n)) = a(n)/A062759(n) > 1.

A071625(a(n)) > 1. - Michael S. Branicky, Sep 01 2022

Example

440 is in the sequence because 440 = 2^3*5*11 and it has two distinct exponents, 3 and 1.

Prog

(PARI) is(n)=#Set(factor(n)[, 2])>1 \\ Charles R Greathouse IV, Sep 18 2015
(Python)
from sympy import factorint
def ok(n): return len(set(factorint(n).values())) > 1
print([k for k in range(190) if ok(k)]) # Michael S. Branicky, Sep 01 2022
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A059404(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def f(x): return n+1-(y:=x.bit_length())+sum(g(integer_nthroot(x, k)[0]) for k in range(1, y))
    kmin, kmax = 1, 2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

Crossrefs

Cf. A003557, A007947, A051904, A062759, A062760, A071625.

Sequence in context: A317711 A359890 A323055 * A303946 A360246 A242416

Adjacent sequences: A059401 A059402 A059403 * A059405 A059406 A059407

Keyword

nonn,easy,changed

Author

Labos Elemer, Jul 18 2001

Status

approved