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

1,1

Links

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

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A085971 A175082 A007916 * A341646 A109421 A335433

Adjacent sequences: A052482 A052483 A052484 * A052486 A052487 A052488

Keyword

nonn,easy

Author

Henry Bottomley, Mar 16 2000

Status

approved