A253641 -id:A253641 - cp's OEIS frontend

A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

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

Offset: 1

Eric W. Weisstein

nonn

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

			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.

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

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.

a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012

# See link.
#
a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):
seq(a(n), n=1..120);  # Alois P. Heinz, Oct 20 2019

Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012

from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022

(define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017

a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 07 2017

More terms from Labos Elemer, Jun 17 2002

A072103 Sorted perfect powers a^b for a, b > 1 with duplication.

Original entry on oeis.org

4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, 81, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 256, 256, 289, 324, 343, 361, 400, 441, 484, 512, 512, 529, 576, 625, 625, 676, 729, 729, 729, 784, 841, 900, 961, 1000, 1024, 1024, 1024, 1089

Offset: 1

Eric W. Weisstein, Jun 18 2002

nonn

If b is the largest integer such that n=a^b for some a > 1, then n occurs d(b)-1 times in this sequence (where d = A000005 is the number of divisors function). (This includes the case where b=1 and n does not occur in the sequence.) - M. F. Hasler, Jan 25 2015

			(a,b) = (2,4) and (4,2) both yield 2^4 = 4^2 = 16, therefore 16 is listed twice.
Similarly, 64 is listed 3 times since (a,b) = (2,6), (4,3) and (8,2) all yield 64.

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999, recomputed with new offset by _M. F. Hasler_, Jan 25 2015
Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A000005, A253641, A253642.

import Data.Set (singleton, findMin, deleteMin, insert)
a072103 n = a072103_list !! (n-1)
a072103_list = f 9 3 $ Set.singleton (4,2) where
   f zz z s
     | xx < zz   = xx : f zz z (Set.insert (x*xx, x) $ Set.deleteMin s)
     | otherwise = zz : f (zz+2*z+1) (z+1) (Set.insert (z*zz, z) s)
     where (xx, x) = Set.findMin s
-- Reinhard Zumkeller, Oct 04 2012

N:= 2000: # to get all entries <= N
sort([seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))]); # Robert Israel, Jan 25 2015

nn=60;Take[Sort[#[[1]]^#[[2]]&/@Tuples[Range[2,nn],2]],nn] (* Harvey P. Dale, Oct 03 2012 *)

is_A072103(n)=ispower(n)
for(n=1,999,(e=ispower(n))||next;fordiv(e,d,d>1 && print1(n","))) \\ M. F. Hasler, Jan 25 2015

import numpy
from math import isqrt
upto = 1090
A072103 = []
for m in range(2,isqrt(upto)+1):
    k = 2
    while m**k < upto:
        A072103.append(m**k)
        k += 1
print(sorted(A072103)) # Karl-Heinz Hofmann, Sep 16 2023

Sum_{i>=2} Sum_{j>=2} 1/i^j = 1.

Offset corrected and examples added by M. F. Hasler, Jan 25 2015

A175064 a(1) = 1; for n >= 2, a(n) = number of ways h to write the n-th perfect power A001597(n) as m^k with m >= 2 and k >= 1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

Perfect powers with first occurrence of h >= 2: 4, 16, 64, 65536, 4096, ... [The perfect power corresponding to h is A175065(h) = 2^A005179(h). - Jianing Song, Oct 27 2024]

			For n = 11: A001597(11) = 64; there are 4 ways to write 64 as m^k: 64^1 = 8^2 = 4^3 = 2^6.

Cf. A253641, A253642, A000005, A001597.

from math import gcd
from sympy import mobius, integer_nthroot, divisor_count, factorint
def A175064(n):
    if n == 1: return 1
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) 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 divisor_count(gcd(*factorint(kmax).values())) # Chai Wah Wu, Aug 13 2024

a(n) = A000005(A253641(A001597(n))) = A253642(n)+1. - M. F. Hasler, Jan 25 2015

Extended by T. D. Noe, Apr 21 2011

Definition clarified by Jonathan Sondow, Nov 30 2012

A253642 Number of ways the perfect power A001597(n) can be written as a^b, with a, b > 1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 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, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

M. F. Hasler, Jan 25 2015

nonn

Run lengths of A072103. Also, the terms a(n) which exceed 1 constitute A175066. - Andrey Zabolotskiy, Aug 17 2016

			a(1)=0 since A001597(1)=1 can be written as a^b for a=1 and any b, but not using a base a > 1.
a(2)=a(3)=a(4)=1 since the following terms 4=2^2, 8=2^3 and 9=3^2 can be written as perfect powers in only one way.
a(5)=2 since A001597(5)=16=a^b for (a,b)=(2,4) and (4,2).

Cf. A001597, A072103, A175064, A253641.

for(n=1,9999,(e=ispower(n))&&print1(numdiv(e)-1,","))

from math import gcd
from sympy import mobius, integer_nthroot, divisor_count, factorint
def A253642(n):
    if n == 1: return 0
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) 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 divisor_count(gcd(*factorint(kmax).values()))-1 # Chai Wah Wu, Aug 13 2024

a(n) = A000005(A253641(A001597(n))) - 1.

a(n) = A175064(n) - 1.

A175067 a(n) is the sum of perfect divisors of n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 12, 13, 14, 15, 22, 17, 18, 19, 20, 21, 22, 23, 24, 30, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 42, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 56, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 78, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

a(n) > n for perfect powers n = A001597(m) for m > 2.

			For n = 8: a(8) = 10; there are two perfect divisors of 8: 2 and 8; their sum is 10.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A175070, A007916, A175068, A253641, A052410.

A175067:= proc(n) local a, d, k; if n=1 then return 1 end if; a:=0; for d in numtheory[divisors](n) minus {1} do for k do if d^k=n then a:= a+d end if; if n <= d^k then break; end if; end do; end do; a end proc:
seq(A175067(n), n=1..80); # Ridouane Oudra, Dec 12 2024
# second Maple program:
a:= n-> add(`if`(n=d^ilog[d](n), d, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..72);  # Alois P. Heinz, Dec 12 2024

Table[Plus @@ (n^(1/Divisors[GCD @@ FactorInteger[n][[All, 2]]])), {n, 72}] (* Ivan Neretin, May 13 2015 *)

a(n) = A175070(n) + n. [Jaroslav Krizek, Jan 24 2010]
From Ridouane Oudra, Dec 12 2024: (Start)
a(n) = n, for n in A007916.
a(n^m) = Sum_{d|m} n^d, for n in A007916 and m an integer >0.
More generally, for all integers n we have :
a(n) = Sum_{d|A253641(n)} n^(d/A253641(n)).
a(n) = Sum_{d|A253641(n)} A052410(n)^d. (End)

Name edited by Michel Marcus, Jun 13 2018

A175068 a(n) = product of perfect divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 12, 13, 14, 15, 128, 17, 18, 19, 20, 21, 22, 23, 24, 125, 26, 81, 28, 29, 30, 31, 64, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 343, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4096, 65, 66, 67, 68, 69, 70

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

A perfect divisor d of n is a divisor such that d^k = n for some k >= 1.

			For n = 8: a(8) = 16; there are two perfect divisors of 8: 2 and 8; their product is 16.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 1000 terms from Harvey P. Dale)
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum 2 (1951), 254-260. See omega(n).

Cf. A001597, A175069, A175084, A175085, A175087.

Cf. A000203, A346403, A253641.

A175068 := proc(n) local a,d,k ; if n = 1 then return 1; end if; a := 1 ; for d in numtheory[divisors](n) minus {1} do for k from 1 do if d^k = n then a := a*d ; end if; if d^k >= n then break; end if; end do: end do: a ; end proc:
seq(A175068(n),n=1..80) ; # R. J. Mathar, Apr 14 2011

Table[Times@@Select[Rest[Divisors[n]],IntegerQ[Log[#,n]]&],{n,70}] (* Harvey P. Dale, May 01 2017 *)

A175068(n) = { my(m=1); fordiv(n,d,if((1==d)||(d^valuation(n,d))==n,m*=d)); (m); }; \\ Antti Karttunen, Nov 21 2017

a(n) > n for perfect powers n = A001597(m) for m > 2.
a(n) = A175069(n) * n. - Jaroslav Krizek, Jan 24 2010
From Ridouane Oudra, Nov 23 2024: (Start)
a(n) = n, for n in A007916.
a(n^m) = n^sigma(m), for n in A007916 and m an integer.
More generally, for all integer n we have :
a(n) = n^(sigma(A253641(n))/A253641(n)).
a(n) = n^(A346403(n)/A253641(n)). (End)

A259362 a(1) = 1, for n > 1: a(n) is the number of ways to write n as a nontrivial perfect power.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Doug Bell, Jun 24 2015

nonn

a(n) = number of integer pairs (i,j) for distinct values of i where i > 0, j > 1 and n = i^j. Since 1 = 1^r for all real values of r, the requirement for a distinct i causes a(1) = 1 instead of a(1) = infinity.
Alternatively, the sequence can be defined as: a(1) = 1, for n > 1: a(n) = number of pairs (i,j) such that i > 0, j > 1 and n = i^j.
A007916 = n, where a(n) = 0.
A001597 = n, where a(n) > 0.
A175082 = n, where n = 1 or a(n) = 0.
A117453 = n, where n = 1 or a(n) > 1.
A175065 = n, where n > 1 and a(n) > 0 and this is the first occurrence in this sequence of a(n).
A072103 = n repeated a(n) times where n > 1.
A075802 = min(1, a(n)).
A175066 = a(n), where n = 1 or a(n) > 1. This sequence is an expansion of A175066.
A253642 = 0 followed by a(n), where n > 1 and a(n) > 0.
A175064 = a(1) followed by a(n) + 1, where n > 1 and a(n) > 0.
Where n > 1, A001597(x) = n (which implies a(n) > 0), i = A025478(x) and j = A253641(n), then a(n) = A000005(j) - 1, which is the number of factors of j greater than 1. The integer pair (i,j) comprises the smallest value i and the largest value j where i > 0, j > 1 and n = i^j. The a(n) pairs of (a,b) where a > 0, b > 1 and n = a^b are formed with b = each of the a(n) factors of j greater than 1. Examples for n = {8,4096}:
  a(8) = 1, A001597(3) = 8, A025478(3) = 2, A253641(8) = 3, 8 = 2^3 and A000005(3) - 1 = 1 because there is one factor of 3 greater than 1 [3]. The set of pairs (a,b) is {(2,3)}.
  a(4096) = 5, A001597(82) = 4096, A025478(82) = 2, A253641(4096) = 12, 4096 = 2^12 and A000005(12) - 1 = 5 because there are five factors of 12 greater than 1 [2,3,4,6,12]. The set of pairs (a,b) is {(64,2),(16,3),(8,4),(4,6),(2,12)}.
A023055 = the ordered list of x+1 with duplicates removed, where x is the number of consecutive zeros appearing in this sequence between any two nonzero terms.
A070428(x) = number of terms a(n) > 0 where n <= 10^x.
a(n) <= A188585(n).

			a(6) = 0 because there is no way to write 6 as a nontrivial perfect power.
a(9) = 1 because there is one way to write 9 as a nontrivial perfect power: 3^2.
a(16) = 2 because there are two ways to write 16 as a nontrivial perfect power: 2^4, 4^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:       1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Squares: 1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:   1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
...
Sum:    oo, 0, 0, 1, 0, 0, 0, 1, 1, ...
a(1)=1:  1, 0, 0, 1, 0, 0, 0, 1, 1, ... (= this sequence). (End)

Doug Bell, Table of n, a(n) for n = 1..5000

Cf. A001597, A007916, A023055, A025478, A070428, A072103, A075090, A075109, A075802, A117453, A175064, A175066, A175082, A188585, A253641, A253642.

Cf. A010052, A010057, A374016, A089361, A089723, A382691 (alternating).

a[n_] := If[n == 1, 1, Sum[Boole[IntegerQ[n^(1/k)]], {k, 2, Floor[Log[2, n]]}]]; Array[a, 100] (* Friedjof Tellkamp, Jun 14 2025 *)
a[n_] := If[n == 1, 1, DivisorSigma[0, Apply[GCD, Transpose[FactorInteger[n]][[2]]]] - 1]; Array[a, 100] (* Michael Shamos, Jul 06 2025 *)

a(n) = if (n==1, 1, sum(i=2, logint(n, 2), ispower(n, i))); \\ Michel Marcus, Apr 11 2025

a(1) = 1, for n > 1: a(n) = A000005(A253641(n)) - 1.
If n not in A001597, then a(n) = 0, otherwise a(n) = A175064(x) - 1 where A001597(x) = n.
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = A089723(n) - 1, for n > 1.
a(n) = A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
Sum_{k>=2..n} a(k) = A089361(n), for n > 1.
G.f.: x + Sum_{j>=2, k>=2} x^(j^k).
Dirichlet g.f.: 1 + Sum_{k>=2} zeta(k*s)-1. (End)

A346403 a(1)=1; for n>1, a(n) gives the sum of the exponents in the different ways to write n as n = x^y, 2 <= x, 1 <= y.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 15 2021

nonn

Denoted by tau(n) in Mycielski (1951), Fehér et al. (2006), and Awel and Küçükaslan (2020).

This function depends only on the prime signature of n (see the Formula section).

			4 = 2^2, gcd(2) = 2, sigma(2) = 3, so a(4) = 3. The representations are 4^1 and 2^2, and 1 + 2 = 3.
144 = 2^4 * 3^2, gcd(4,2) = 2, sigma(2) = 3, so a(144) = 3. The representations are 144^1 and 12^2, and 1 + 2 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Abdu Awel and M. Küçükaslan, A Note on Statistical Limit and Cluster Points of the Arithmetical Functions a_p(n), gamma(n), and tau(n) in the Sense of Density, Journal of the Indonesian Mathematical Society, Vol. 26, No. 2 (2020), pp. 224-233.
Zoltán Fehér, Béla László, Martin Mačaj and Tibor Šalát, Remarks on arithmetical functions a_p(n), gamma(n), tau(n), Annales Mathematicae et Informaticae, Vol. 33 (2006), pp. 35-43.
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260.

Cf. A000203, A013661, A089723, A052409, A253641.

A253641:=proc(n) if n in {0,1} then 1 else igcd(map(i->i[2], ifactors(n)[2])[]); fi; end: seq(numtheory[sigma](A253641(n)), n=1..120); # Ridouane Oudra, Jun 04 2025

a[n_] := DivisorSigma[1, GCD @@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

a(n) = if (n==1, 1, sigma(gcd(factor(n)[,2]))); \\ Michel Marcus, Jul 16 2021

If n = Product_{i} p_i^e_i, then a(n) = sigma(gcd()).
Sum_{n>=1} (a(n)-1)/n = Pi^2/6 + 1 (= A013661 + 1) (Mycielski, 1951).
a(n) = sigma(A052409(n)), for n>1. - Ridouane Oudra, Nov 23 2024
a(n) = sigma(A253641(n)). - Ridouane Oudra, Jun 04 2025

cp's OEIS Frontend

A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

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A072103 Sorted perfect powers a^b for a, b > 1 with duplication.

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A175064 a(1) = 1; for n >= 2, a(n) = number of ways h to write the n-th perfect power A001597(n) as m^k with m >= 2 and k >= 1.

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A253642 Number of ways the perfect power A001597(n) can be written as a^b, with a, b > 1.

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A175067 a(n) is the sum of perfect divisors of n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

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A175068 a(n) = product of perfect divisors of n.

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