[go: up one dir, main page]

login
A072774
Powers of squarefree numbers.
124
1, 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
OFFSET
1,2
COMMENTS
a(n) = A072775(n)^A072776(n); complement of A059404.
Essentially the same as A062770. - R. J. Mathar, Sep 25 2008
Numbers m such that in canonical prime factorization all prime exponents are identical: A124010(m,k) = A124010(m,1) for k = 2..A000005(m). - Reinhard Zumkeller, Apr 06 2014
Heinz numbers of uniform partitions. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 16 2018
LINKS
MATHEMATICA
Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)
PROG
(Haskell)
import Data.Map (empty, findMin, deleteMin, insert)
import qualified Data.Map.Lazy as Map (null)
a072774 n = a072774_list !! (n-1)
(a072774_list, a072775_list, a072776_list) = unzip3 $
(1, 1, 1) : f (tail a005117_list) empty where
f vs'@(v:vs) m
| Map.null m || xx > v = (v, v, 1) :
f vs (insert (v^2) (v, 2) m)
| otherwise = (xx, bx, ex) :
f vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014
(PARI) is(n)=ispower(n, , &n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A072774(n):
def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))-1
def f(x): return n-2+x-sum(g(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length()))
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
Cf. A072777 (subsequence), A005117, A072778, A329332 (tabular arrangement).
A subsequence of A242414.
Sequence in context: A242414 A360249 A360247 * A062770 A359889 A236510
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 10 2002
STATUS
approved