A052485 Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

Author

Henry Bottomley, Mar 16 2000

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694 (complement), A112526. Not the same as A007916.

Programs

Haskell

a052485 n = a052485_list !! (n-1)
a052485_list = filter ((== 0) . a112526) [1..]
-- Reinhard Zumkeller, Sep 16 2011

Mathematica

Select[Range[1000], Min[FactorInteger[#][[All, 2]]] <= 1 &] (* Geoffrey Critzer, Feb 11 2015 *)

PARI

is(n)=n>1 && vecmin(factor(n)[,2])==1 \\ Charles R Greathouse IV, Mar 19 2014

PARI

is(n)=!ispowerful(n) \\ Charles R Greathouse IV, Sep 18 2015

Python

from math import isqrt
from sympy import integer_nthroot, factorint
def A052485(n):
    def f(x): return int(n+sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 10 2024

Formula

A112526(a(n)) = 0. - Reinhard Zumkeller, Sep 16 2011

a(n) ~ n. - Charles R Greathouse IV, Jul 19 2012

a(n) = n + O(sqrt(n)). - Charles R Greathouse IV, Jul 08 2022

Sum_{n>=1} 1/a(n)^s = zeta(s) - zeta(2*s)*zeta(3*s)/zeta(6*s), for s > 1. - Amiram Eldar, May 13 2023