login
A182853
Squarefree composite integers and powers of squarefree composite integers.
20
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.
LINKS
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map
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.
Sequence in context: A289619 A329140 A362605 * A212168 A344585 A080365
KEYWORD
nonn,easy
AUTHOR
Matthew Vandermast, Jan 04 2011
STATUS
approved