A062770 n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Author

Labos Elemer, Jul 18 2001

Keywords

• nonn

Comments

The sequence contains numbers m such that the exponents e are identical for all prime power factors p^e | m. It is clear from this alternate definition that m / K^E = 1 iff E is an integer. - Michael De Vlieger, Jun 24 2022

Examples

			Primes, squarefree numbers and perfect powers are here.
From _Michael De Vlieger_, Jun 24 2022 (Start):
144 cannot be in the sequence, since the exponents of its prime power factors differ. The squarefree kernel of 144 = 2^4 * 3^2 is 2*3 = 6. The largest power of 6 less than 144 is 36. 144/36 = 4, so it is not in the sequence.
216 is in the sequence because 216 = 2^3 * 3^3 is 2*3 = 6. But 216 = 6^3, hence 6^3 / 6^3 = 1. (End)
		

Links

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

Crossrefs

Cf. A062759, A062760, A007947, A003557, A051904, A005117, A001597, A072774.

Programs

Mathematica

Select[Range[2, 2^16], Length@ Union@ FactorInteger[#][[All, -1]] == 1 &] (* Michael De Vlieger, Jun 24 2022 *)

PARI

is(n)=ispower(n,,&n); issquarefree(n) && n>1 \\ Charles R Greathouse IV, Sep 18 2015

PARI

is(n)=#Set(factor(n)[,2])==1 \\ Charles R Greathouse IV, Sep 18 2015

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A062770(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-2+x+(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

Formula

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

a(n) = A072774(n+1). - Chai Wah Wu, Aug 19 2024

Extensions

Offset corrected by Charles R Greathouse IV, Sep 18 2015