A182853 Squarefree composite integers and powers of squarefree composite integers.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset

1,1

Comments

Numbers that require exactly three iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.

Numbers such that A001221(n) > 1 and A071625(n) = 1.

Union of A120944 and A303606. - Michael De Vlieger, Oct 02 2025

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Fixed Point

Eric Weisstein's World of Mathematics, Map

Mathematica

nn = 180; i = k = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range[nn], And[CompositeQ[#], SquareFreeQ[#]] &]]; Union@ Reap[While[j = 1; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 1, k++; i++] ][[-1, 1]] (* Michael De Vlieger, Oct 02 2025 *)

Prog

(Scheme)
(define A182853 (MATCHING-POS 1 1 (lambda (n) (= 3 (A182850 n))))) ;; After the alternative definition of the sequence given by the original author. Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 05 2016
(PARI) isoka(n) = (omega(n) > 1) && issquarefree(n); \\ A120944
isok(n) = isoka(n) || (ispower(n, , &k) && isoka(k)); \\ Michel Marcus, Jun 24 2017
(Python)
from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A182853(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    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

Crossrefs

Numbers n such that A182850(n) = 3. See also A182854, A182855.

Subsequence of A072774 and A182851.

Cf. A120944, A303606.

Sequence in context: A289619 A329140 A362605 * A212168 A344585 A080365

Adjacent sequences: A182850 A182851 A182852 * A182854 A182855 A182856

Keyword

nonn,easy

Author

Matthew Vandermast, Jan 04 2011

Status

approved