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Search: a072776 -id:a072776
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A052409
a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).
+10
125
0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Greatest common divisor of all prime-exponents in canonical factorization of n for n>1: a(n)>1 iff n is a perfect power; a(A001597(k))=A025479(k). - Reinhard Zumkeller, Oct 13 2002
a(1) set to 0 since there is no largest finite integer power m for which a representation of the form 1 = 1^m exists (infinite largest m). - Daniel Forgues, Mar 06 2009
A052410(n)^a(n) = n. - Reinhard Zumkeller, Apr 06 2014
Positions of 1's are A007916. Smallest base is given by A052410. - Gus Wiseman, Jun 09 2020
LINKS
Daniel Forgues, Table of n, a(n) for n = 1..100000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
Eric Weisstein's World of Mathematics, Power
Eric Weisstein's World of Mathematics, Perfect Power
Index entries for sequences computed from exponents in factorization of n
FORMULA
a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 07 2017
EXAMPLE
n = 72 = 2*2*2*3*3: GCD[exponents] = GCD[3,2] = 1. This is the least n for which a(n) <> A051904(n), the minimum of exponents.
For n = 10800 = 2^4 * 3^3 * 5^2, GCD[4,3,2] = 1, thus a(10800) = 1.
MAPLE
# See link.
#
a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):
seq(a(n), n=1..120); # Alois P. Heinz, Oct 20 2019
MATHEMATICA
Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* Ray Chandler, Jan 24 2006 *)
PROG
(Haskell)
a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012
(PARI) a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012
(Scheme) (define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017
(Python)
from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022
CROSSREFS
Cf. A052410, A005361, A051903, A072411-A072414, A124010, A075802, A072776, A270492.
Apart from the initial term essentially the same as A253641.
Differs from A051904 for the first time at n=72, where a(72) = 1, while A051904(72) = 2.
Differs from A158378 for the first time at n=10800, where a(10800) = 1, while A158378(10800) = 2.
Cf. A000005, A000961, A001597, A052410, A303386, A327501.
KEYWORD
nonn
AUTHOR
Eric W. Weisstein
EXTENSIONS
More terms from Labos Elemer, Jun 17 2002
STATUS
approved
A072774
Powers of squarefree numbers.
+10
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
(list; graph; refs; listen; history; text; internal format)
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
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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.
Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 10 2002
STATUS
approved
A072777
Powers of squarefree numbers that are not squarefree.
+10
14
4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 169, 196, 216, 225, 243, 256, 289, 343, 361, 441, 484, 512, 529, 625, 676, 729, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
For all n exists k: a(n) = A072774(k) and A072776(k) > 1.
Numbers k such that every prime in the prime factorization of k is raised to the same power > 1; k is a term iff k/A007947(k)^m = 1 for some m > 1. - David James Sycamore, Jun 12 2024
LINKS
Stanislav Sykora and Reinhard Zumkeller, Table of n, a(n) for n = 1..20000 (first 10000 terms from Reinhard Zumkeller)
FORMULA
Sum_{n>=1} 1/a(n) = Sum_{n>=2} mu(n)^2/(n*(n-1)) = Sum_{n>=2} (zeta(n)/zeta(2*n) - 1) = 0.8486338679... (A368250). - Amiram Eldar, Jul 22 2020
EXAMPLE
The number 144 = 12^2 is not a member because 12 is not squarefree.
64 = 2^6 and 49 = 7^2 are members because, though not squarefree, they are powers of the squarefree numbers 2 and 7, respectively. Note that 64 is included even though it is also a square of a nonsquarefree number. - Stanislav Sykora, Jul 11 2014
MATHEMATICA
Select[Range[2000], Length[u = Union[FactorInteger[#][[All, 2]]]] == 1 && u[[1]] > 1 &] (* Jean-François Alcover, Mar 27 2013 *)
PROG
(Haskell)
import Data.Map (singleton, findMin, deleteMin, insert)
a072777 n = a072777_list !! (n-1)
a072777_list = f 9 (drop 2 a005117_list) (singleton 4 (2, 2)) where
f vv vs'@(v:ws@(w:_)) m
| xx < vv = xx : f vv vs' (insert (bx*xx) (bx, ex+1) $ deleteMin m)
| xx > vv = vv : f (w*w) ws (insert (v^3) (v, 3) m)
where (xx, (bx, ex)) = findMin m
-- Reinhard Zumkeller, Apr 06 2014
(PARI) BelongsToA(n) = {my(f, k, e); if(n == 1, return(0));
f = factor(n); e = f[1, 2]; if(e == 1, return(0));
for(k = 2, #f[, 2], if(f[k, 2] != e, return(0))); return(1); }
Ntest(nmax, test) = {my(k = 1, n = 0, v); v = vector(nmax); while(1, n++; if(test(n), v[k] = n; k++; if(k > nmax, break)); ); return(v); }
a = Ntest(20000, BelongsToA) \\ Note: not very efficient. - Stanislav Sykora, Jul 11 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 A072777(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-1+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, 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. A013929, A368250.
Cf. A005117, subsequence of A001597 and A072774.
Cf. A007947.
KEYWORD
nonn,nice
AUTHOR
Reinhard Zumkeller, Jul 10 2002
STATUS
approved
A072775
Squarefree kernels of powers of squarefree numbers.
+10
4
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 13, 14, 15, 2, 17, 19, 21, 22, 23, 5, 26, 3, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 41, 42, 43, 46, 47, 7, 51, 53, 55, 57, 58, 59, 61, 62, 2, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 3, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 10, 101
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) = A007947(A072774(n));
A072774(n) = a(n)^A072776(n);
A072774(n) is squarefree iff A072774(n)=a(n).
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
PROG
(Haskell)
a072775 n = a072775_list !! (n-1) -- a072775_list defined in A072774.
-- Reinhard Zumkeller, Apr 06 2014
(Python)
from math import isqrt, prod
from sympy import mobius, integer_nthroot, primefactors
def A072775(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 prod(primefactors(kmax)) # Chai Wah Wu, Aug 19 2024
CROSSREFS
Cf. A052410.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 10 2002
STATUS
approved
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