A278028 -id:A278028 - cp's OEIS frontend

A288636 Height of power-tower factorization of n. Row lengths of A278028.

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

Offset: 1

Gus Wiseman, Jun 12 2017

nonn

After a(1)=0 this sequence has many terms equal to A089723. It first differs at a(64)=2, A089723(64)=4.

First positions of a(n) = {0, 1, 2, 3, 4} are n = {1, 2, 4, 16, 65536}. - Michael De Vlieger, Nov 24 2017

Michael De Vlieger, Table of n, a(n) for n = 1..65536

Cf. A007916, A089723, A164337, A277562, A277564, A278028.

a[n_] := If[n == 1, 0, 1+a[GCD @@ FactorInteger[n][[All, 2]]]];
Array[a,100]

A007916 Numbers that are not perfect powers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

R. Muller

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1-to-1 correspondence between integers N >= 2 and sequences a(x_1),a(x_2),...,a(x_k) of terms from this sequence. Every N >= 2 can be written uniquely as a "power tower"
N = a(x_1)^a(x_2)^a(x_3)^...^a(x_k),
where the exponents are to be nested from the right.
Proof: If N is not a perfect power then N = a(x) for some x, and we are done. Otherwise, write N = a(x_1)^M for some M >=2, and repeat the process. QED
Of course, prime numbers also have distinct power towers (see A164336). (End)
These numbers can be computed with a modified Sieve of Eratosthenes: (1) start at n=2; (2) if n is not crossed out, then append n to the sequence and cross out all powers of n; (3) set n = n+1 and go to step 2. - Sam Alexander, Dec 15 2003
These are all numbers such that the multiplicities of the prime factors have no common divisor. The first number in the sequence whose prime multiplicities are not coprime is 180 = 2 * 2 * 3 * 3 * 5. Mathematica: CoprimeQ[2,2,1]->False. - Gus Wiseman, Jan 14 2017

			Example of the power tower factorizations for the first nine positive integers: 1=1, 2=a(1), 3=a(2), 4=a(1)^a(1), 5=a(3), 6=a(4), 7=a(5), 8=a(1)^a(2), 9=a(2)^a(1). - _Gus Wiseman_, Oct 20 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
Joakim Munkhammar, The Riemann zeta function as a sum of geometric series, The Mathematical Gazette (2020) Vol. 104, Issue 561, 527-530.
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993
Index entries for sequences generated by sieves

Complement of A001597. Union of A052485 and A052486.
Cf. A144338, A277562, A277564, A075802.
Cf. A153158 (squares of these numbers).
See A277562, A277564, A277576, A277615 for more about the power towers.
A278029 is a left inverse.
Cf. A052409.

a007916 n = a007916_list !! (n-1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

[n : n in [2..1000] | not IsPower(n) ];

Maple
```
See link.
```

a = {}; Do[If[Apply[GCD, Transpose[FactorInteger[n]][[2]]] == 1, a = Append[a, n]], {n, 2, 200}];
Select[Range[2,200],GCD@@FactorInteger[#][[All,-1]]===1&] (* Michael De Vlieger, Oct 21 2016. Corrected by Gus Wiseman, Jan 14 2017 *)

is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import mobius, integer_nthroot
def A007916(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 13 2024

A075802(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2009
Gcd(exponents in prime factorization of a(n)) = 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n. - Charles R Greathouse IV, Jul 01 2013
A052409(a(n)) = 1. - Ridouane Oudra, Nov 23 2024

More terms from Henry Bottomley, Sep 12 2000
Edited by Charles R Greathouse IV, Mar 18 2010
Further edited by N. J. A. Sloane, Nov 09 2016

A052410 Write n = m^k with m, k integers, k >= 1, then a(n) is the smallest possible choice for m.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Eric W. Weisstein

nonn

Value of m in m^p = n, where p is the largest possible power (see A052409).
For n > 1, n is a perfect power iff a(n) <> n. - Reinhard Zumkeller, Oct 13 2002
a(n)^A052409(n) = n. - Reinhard Zumkeller, Apr 06 2014
Every integer root of n is a power of a(n). All entries (except 1) belong to A007916. - Gus Wiseman, Sep 11 2017

Daniel Forgues, Table of n, a(n) for n=1..100000
Eric Weisstein's World of Mathematics, Power
Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A025478, A007916, A027748, A052409, A072775, A124010, A175781, A278028, A288636, A289023.

a052410 n = product $ zipWith (^)
                      (a027748_row n) (map (`div` (foldl1 gcd es)) es)
            where es = a124010_row n
-- Reinhard Zumkeller, Jul 15 2012

a:= n-> (l-> (t-> mul(i[1]^(i[2]/t), i=l))(
         igcd(seq(i[2], i=l))))(ifactors(n)[2]):
seq(a(n), n=1..74);  # Alois P. Heinz, Jul 22 2024

Table[If[n==1, 1, n^(1/(GCD@@(Last/@FactorInteger[n])))], {n, 100}]

a(n) = if (ispower(n,,&r), r, n); \\ Michel Marcus, Jul 19 2017

def upto(n):
    list = [1] + [0] * (n - 1)
    for i in range(2, n + 1):
        if not list[i - 1]:
            j = i
            while j <= n:
                list[j - 1] = i
                j *= i
    return list
# M. Eren Kesim, Jun 03 2021

from math import gcd
from sympy import integer_nthroot, factorint
def A052410(n): return integer_nthroot(n,gcd(*factorint(n).values()))[0] if n>1 else 1 # Chai Wah Wu, Mar 02 2024

a(A001597(k)) = A025478(k).

a(n) = A007916(A278028(n,1)). - Gus Wiseman, Sep 11 2017

Definition edited (in a complementary form to A052409) by Daniel Forgues, Mar 14 2009
Corrected by Charles R Greathouse IV, Sep 02 2009
Definition edited by N. J. A. Sloane, Sep 03 2010

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

Original entry on oeis.org

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

A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

Original entry on oeis.org

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

Offset: 1

Naohiro Nomoto, Jan 07 2004

This function depends only on the prime signature of n. - Franklin T. Adams-Watters, Mar 10 2006
a(n) is the number of perfect divisors of n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) > 1 for perfect powers n = A001597(m) for m > 2. - Jaroslav Krizek, Jan 23 2010
Also the number of uniform perfect integer partitions of n - 1. An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset. The Heinz numbers of these partitions are given by A326037. The a(16) = 3 partitions are: (8,4,2,1), (4,4,4,1,1,1), (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1). - Gus Wiseman, Jun 07 2019
The record values occur at 1 and at 2^A002182(n) for n > 1. - Amiram Eldar, Nov 06 2020

			144 = 2^4 * 3^2, gcd(4,2) = 2, d(2) = 2, so a(144) = 2. The representations are 144^1 and 12^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:          1, 2, 3, 4, 5, 6, 7, 8, 9, ...
----------------------------------------------------
1st powers: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012)
Squares:    1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:      1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
Quartics:   1, 0, 0, 0, 0, 0, 0, 0, 0, ... (A374016)
...
Sum:       oo, 1, 1, 2, 1, 1, 1, 2, 2, ...
a(1)=1:     1, 1, 1, 2, 1, 1, 1, 2, 2, ... (= this sequence). (End)

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, A new arithmetic function of combinatorial significance, J. Number Theory, Vol. 5, No. 3 (1973), pp. 218-223. 1973JNT.....5..218G
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260. See gamma(n).
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A000005, A277564, A278028.
Cf. A000961, A002033, A002182, A007916, A047966, A070941, A108917, A126796, A276024, A326035, A326036, A326037.
Cf. A010052, A010057, A374016, A259362.

with(numtheory):
A089723 := proc(n) local t1,t2,g,j;
if n=1 then 1 else
t1:=ifactors(n)[2]; t2:=nops(t1); g := t1[1][2];
for j from 2 to t2 do g:=gcd(g,t1[j][2]); od:
tau(g); fi; end;
[seq(A089723(n),n=1..100)]; # N. J. A. Sloane, Nov 10 2016

Table[DivisorSigma[0, GCD @@ FactorInteger[n][[All, 2]]], {n, 100}] (* Gus Wiseman, Jun 12 2017 *)

a(n) = if (n==1, 1, numdiv(gcd(factor(n)[,2]))); \\ Michel Marcus, Jun 13 2017

from math import gcd
from sympy import factorint, divisor_sigma
def a(n):
    if n == 1: return 1
    e = list(factorint(n).values())
    g = e[0]
    for ei in e[1:]: g = gcd(g, ei)
    return divisor_sigma(g, 0)
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 15 2021

If n = Product p_i^e_i, a(n) = d(gcd()). - Franklin T. Adams-Watters, Mar 10 2006
Sum_{n=1..m} a(n) = A255165(m) + 1. - Richard R. Forberg, Feb 16 2015
Sum_{n>=2} a(n)/n^s = Sum_{n>=2} 1/(n^s-1) = Sum_{k>=1} (zeta(s*k)-1) for all real s with Re(s) > 1 (Golomb, 1973). - Amiram Eldar, Nov 06 2020
For n > 1, a(n) = Sum_{i=1..floor(n/2)} floor(n^(1/i))-floor((n-1)^(1/i)). - Wesley Ivan Hurt, Dec 08 2020
Sum_{n>=1} (a(n)-1)/n = 1 (Mycielski, 1951). - Amiram Eldar, Jul 15 2021
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = 1 + A259362(n) = 1 + A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
G.f.: x + Sum_{j>=2, k>=1} x^(j^k). (End)

A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 9, 4, 7, 9, 11, 6, 9, 11, 13, 7, 8, 11, 13, 15, 9, 10, 13, 15, 9, 17, 6, 11, 12, 15, 17, 6, 11, 19, 8, 9, 13, 14, 17, 19, 8, 13, 21, 10, 11, 15, 16, 19, 11, 21, 10, 15, 23, 12, 13, 17, 18, 21, 13, 23, 12, 17, 25, 7, 14, 15, 19, 20, 23, 15, 25, 14, 19, 27, 9, 16, 17, 21, 22, 25, 9, 17, 27, 16, 21, 29, 11, 18, 19, 23, 24, 27, 11, 19, 29, 18, 23, 31, 13, 11

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

A free pure symmetric multifunction in one symbol f in PSM(x) is either (case 1) f = the symbol x, or (case 2) f = an expression of the form h[g_1,...,g_k] where h is in PSM(x), each of the g_i for i=1..(k>0) is in PSM(x), and for i < j we have g_i <= g_j under a canonical total ordering of PSM(x), such as the Mathematica ordering of expressions. For a positive integer n we define a free pure symmetric multifunction j(n) by: j(1)=x; j(n>1) = j(h)[j(g_1),...,j(g_k)] where n = r(h)^(p(g_1)*...*p(g_k)-1). Here r(n) is the n-th number that is not a perfect power (A007916) and p(n) is the n-th prime number (A000040). See example. Then a(n) is the number of brackets [...] plus the number of x's in j(n).

			The first 20 free pure symmetric multifunctions in x are:
j(1)  = j(1)            = x
j(2)  = j(1)[j(1)]      = x[x]
j(3)  = j(2)[j(1)]      = x[x][x]
j(4)  = j(1)[j(2)]      = x[x[x]]
j(5)  = j(3)[j(1)]      = x[x][x][x]
j(6)  = j(4)[j(1)]      = x[x[x]][x]
j(7)  = j(5)[j(1)]      = x[x][x][x][x]
j(8)  = j(1)[j(1),j(1)] = x[x,x]
j(9)  = j(2)[j(2)]      = x[x][x[x]]
j(10) = j(6)[j(1)]      = x[x[x]][x][x]
j(11) = j(7)[j(1)]      = x[x][x][x][x][x]
j(12) = j(8)[j(1)]      = x[x,x][x]
j(13) = j(9)[j(1)]      = x[x][x[x]][x]
j(14) = j(10)[j(1)]     = x[x[x]][x][x][x]
j(15) = j(11)[j(1)]     = x[x][x][x][x][x][x]
j(16) = j(1)[j(3)]      = x[x[x][x]]
j(17) = j(12)[j(1)]     = x[x,x][x][x]
j(18) = j(13)[j(1)]     = x[x][x[x]][x][x]
j(19) = j(14)[j(1)]     = x[x[x]][x][x][x][x]
j(20) = j(15)[j(1)]     = x[x][x][x][x][x][x][x].

Cf. A005043, A007916, A106490, A277564, A277615, A277996, A278028, A280000.

Cf. A279984 (numbers j(n)[x]=j(prime(n))), A277576 (numbers j(n)=x[x][x][x]...), A058891 (numbers j(n)=x[x,...,x]), A279969 (numbers j(n)=x[x[...[x]]]).

nn=100;
radQ[n_]:=If[n===1,False,SameQ[GCD@@FactorInteger[n][[All,2]],1]];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Set@@@Array[radPi[rad[#]]==#&,nn];
jfac[n_]:=With[{g=GCD@@FactorInteger[n+1][[All,2]]},JIX[radPi[Power[n+1,1/g]],Flatten[Cases[FactorInteger[g+1],{p_,k_}:>ConstantArray[PrimePi[p],k]]]]];
diwt[n_]:=If[n===1,1,Apply[1+diwt[#1]+Total[diwt/@#2]&,jfac[n-1]]];
Array[diwt,nn]

a(A007916(h)^(A000040(g_1)*...*A000040(g_k)-1)) = 1 + a(h) + a(g_1) + ... + a(g_k).

A277564 Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. The sequence is an irregular triangle read by rows, where the n-th row lists n followed by x_1, ..., x_k.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 1, 5, 3, 6, 4, 7, 5, 8, 1, 2, 9, 2, 1, 10, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 11, 16, 1, 1, 1, 17, 12, 18, 13, 19, 14, 20, 15, 21, 16, 22, 17, 23, 18, 24, 19, 25, 3, 1, 26, 20, 27, 2, 2, 28, 21, 29, 22, 30, 23, 31, 24, 32, 1, 3, 33, 25, 34, 26, 35, 27, 36, 4, 1, 37, 28, 38, 29, 39, 30, 40, 31

Offset: 1

Gus Wiseman, Oct 20 2016

The row lengths are A288636(n) + 1. - Gus Wiseman, Jun 12 2017

See A278028 for a version in which row n simply lists x_1, x_2, ..., x_k (omitting the initial n).

			1 is represented by the empty sequence (), by convention.
Successive rows of the triangle are as follows (c(k) denotes the k-th non-prime-power, A007916(k)):
2, 1,
3, 2,
4, 1, 1,
5, 3,
6, 4, because 6 = c(4)
7, 5,
8, 1, 2, because 8 = 2^3 = c(1)^c(2)
9, 2, 1,
10, 6,
11, 7,
...
16, 1, 1, 1, because 16 = 2^4 = c(1)^4 = c(1)^(c(1)^2) = c[1]^(c[1]^c[1])
17, 12,
...
This sequence represents a bijection N -> Q where Q is the set of all finite sequences of positive integers: 1->(), 2->(1), 3->(2), 4->(1 1), 5->(3), 6->(4), 7->(5), 8->(1 2), 9->(2 1), ...

Gus Wiseman, Table of n, a(n) for n = 1..20131
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A007916, A277562, A164336, A000961, A164337, A278028, A089723.

Maple
```
See link.
```

nn=10000;radicalQ[1]:=False;radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All,2]],1];
hyperfactor[1]:={};hyperfactor[n_?radicalQ]:={n};hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All,2]]},Prepend[hyperfactor[g],Product[Apply[Power[#1,#2/g]&,r],{r,FactorInteger[n]}]]];
rad[0]:=1;rad[n_?Positive]:=rad[n]=NestWhile[#+1&,rad[n-1]+1,Not[radicalQ[#]]&];Set@@@Array[radPi[rad[#]]==#&,nn];
Flatten[Join[{#},radPi/@hyperfactor[#]]&/@Range[nn]]

Edited by N. J. A. Sloane, Nov 09 2016

A289023 Position in the sequence of numbers that are not perfect powers (A007916) of the smallest positive integer x such that for some positive integer y we have n = x^y (A052410).

Original entry on oeis.org

1, 2, 1, 3, 4, 5, 1, 2, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 3, 20, 2, 21, 22, 23, 24, 1, 25, 26, 27, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 5, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 1, 54, 55, 56, 57, 58, 59, 60

Offset: 2

Gus Wiseman, Jun 22 2017

nonn

Every pair p of positive integers is of the form p = (a(n), A052409(n)) for exactly one n.

			a(27)=2 because the smallest root of 27 is 3, and 3 is the 2nd entry of A007916.
a(25)=3 because the smallest root of 25 is 5, and 5 is the 3rd entry of A007916.

Cf. A007916, A052409, A052410, A278028, A288636.

nn=100;
q=Table[Power[n,1/GCD@@FactorInteger[n][[All,2]]],{n,2,nn}];
q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}]

a(n) = if (ispower(n,,&r), x = r, x = n); sum(k=2, x, ispower(k)==0); \\ Michel Marcus, Jul 19 2017

For n>1 we have a(n) = A278028(n,1).

A278029 a(1) = 0; for n > 1, a(n) = k if n is a non-perfect-power, A007916(k); or 0 if n is a perfect power.

Original entry on oeis.org

0, 1, 2, 0, 3, 4, 5, 0, 0, 6, 7, 8, 9, 10, 11, 0, 12, 13, 14, 15, 16, 17, 18, 19, 0, 20, 0, 21, 22, 23, 24, 0, 25, 26, 27, 0, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 0, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 0

Offset: 1

N. J. A. Sloane, Nov 10 2016

Rémy Sigrist, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A007916, A072813, A289023.

FoldList[Boole[#2 != #1] #2 &, #] &@ Accumulate@ Array[Boole[And[# > 1, CoprimeQ @@ FactorInteger[#][[All, -1]]]] &, 81] (* Michael De Vlieger, Dec 18 2016 *)

Name corrected by Peter Munn, Feb 28 2024

A304481 Turn the power-tower for n upside-down.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 26, 27, 28, 29, 30, 31, 25, 33, 34, 35, 64, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 128, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 36, 65, 66, 67

Offset: 1

Gus Wiseman, May 13 2018

This is an involution of the positive integers.

The power-tower for n is defined as follows. Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. Then a(n) = c(x_k)^...^c(x_3)^c(x_2)^c(x_1).

			The power tower of 81 is 3^2^2, which turned upside-down is 2^2^3 = 256, so a(81) = 256.

Robert Israel, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304491, A304492, A304495.

f:= proc(n,r) local F,a,y;
     if n = 1 then return 1 fi;
     F:= ifactors(n)[2];
     y:= igcd(seq(t[2],t=F));
     if y = 1 then return n^r fi;
     a:= mul(t[1]^(t[2]/y),t=F);
     procname(y,a^r)
end proc:
seq(f(n,1),n=1..100); # Robert Israel, May 13 2018

tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[Power@@Reverse[tow[n]],{n,100}]

cp's OEIS Frontend

A288636 Height of power-tower factorization of n. Row lengths of A278028.

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A007916 Numbers that are not perfect powers.

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A052410 Write n = m^k with m, k integers, k >= 1, then a(n) is the smallest possible choice for m.

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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).

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A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

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A279944 Number of positions in the free pure symmetric multifunction in one symbol with j-number n.

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