A289023 -id:A289023 - cp's OEIS frontend

A072774 Powers of squarefree numbers.

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

Reinhard Zumkeller, Jul 10 2002

nonn

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A059404.
Cf. A072775, A072776, A072777 (subsequence), A005117, A072778, A124010, A329332 (tabular arrangement), A384667 (characteristic function).
A subsequence of A242414.
Cf. A000005, A000009, A000837, A007916, A013661, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796, A302979.

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

isA := n -> n=1 or is(1 = nops({seq(p[2], p in ifactors(n)[2])})):
select(isA, [seq(1..97)]);  # Peter Luschny, Jun 10 2025

Select[Range[100], Length[Union[FactorInteger[#][[All, 2]]]] == 1 &] (* Geoffrey Critzer, Mar 30 2015 *)

is(n)=ispower(n,,&n); issquarefree(n) \\ Charles R Greathouse IV, Oct 16 2015

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

a(n) = A072775(n)^A072776(n).
Sum_{n>=1} 1/a(n)^s = 1 + Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025
a(n)/n ~ Pi^2/6 (A013661). - Friedjof Tellkamp, Jun 09 2025

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

A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 6, 1, 1, 2, 2, 1, 7, 1, 2, 8, 1, 3, 2, 2, 1, 1, 9, 1, 2, 3, 1, 1, 2, 1, 4, 10, 1, 1, 1, 11, 1, 2, 2, 1, 1, 3, 3, 1, 1, 12, 1, 1, 2, 2, 1, 3, 13, 1, 1, 1, 14, 1, 5, 2, 3, 1, 1, 15, 1, 2, 4, 1, 3, 2, 2, 1, 6, 16, 1, 2

Offset: 1

Gus Wiseman, Nov 25 2023

Row n = 1 is empty.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.
Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.

			The first 45 rows:
     1: {}      16: {1}       31: {11}
     2: {1}     17: {7}       32: {1}
     3: {2}     18: {1,2}     33: {2,2}
     4: {1}     19: {8}       34: {1,1}
     5: {3}     20: {1,3}     35: {3,3}
     6: {1,1}   21: {2,2}     36: {1,1}
     7: {4}     22: {1,1}     37: {12}
     8: {1}     23: {9}       38: {1,1}
     9: {2}     24: {1,2}     39: {2,2}
    10: {1,1}   25: {3}       40: {1,3}
    11: {5}     26: {1,1}     41: {13}
    12: {1,2}   27: {2}       42: {1,1,1}
    13: {6}     28: {1,4}     43: {14}
    14: {1,1}   29: {10}      44: {1,5}
    15: {2,2}   30: {1,1,1}   45: {2,3}

Indices of empty and singleton rows are A000961.
Row lengths are A001221.
Depends only on rootless base A052410, see A007916.
Row minima are A055396.
Rows have A071625 distinct elements.
Indices of constant rows are A072774.
Indices of strict rows are A130091.
Rows have Heinz numbers A367580.
Row sums are A367581.
Row maxima are A367583, opposite A367587.
Index of first row with Heinz number n is A367584.
Sorted row indices of first appearances are A367585.
Indices of rows of the form {1,1,...} are A367586.
Agrees with sorted prime signature at A367683, counted by A367682.
A submultiset of prime indices at A367685, counted by A367684.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 lists prime multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

For all positive integers n and k, row n^k is the same as row n.

A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 2, 6, 1, 2, 1, 7, 2, 8, 3, 2, 1, 9, 2, 3, 1, 2, 4, 10, 1, 11, 1, 2, 1, 3, 1, 12, 1, 2, 3, 13, 1, 14, 5, 3, 1, 15, 2, 4, 3, 2, 6, 16, 2, 3, 4, 2, 1, 17, 2, 18, 1, 4, 1, 3, 1, 19, 7, 2, 1, 20, 2, 21, 1, 3, 8, 4, 1, 22, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 28 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}.

			For 450 = 2^1 * 3^2 * 5^2, we have MMK({1,2,2,3,3}) = {1,2,2} so a(450) = 2.

Positions of first appearances are A008578.
Depends only on rootless base A052410, see A007916, A052409.
For minimum instead of maximum element we have A055396.
Row maxima of A367579.
Greatest prime index of A367580.
Positions of 1's are A367586 (powers of even squarefree numbers).
The opposite version is A367587.
A007947 gives squarefree kernel.
A072774 lists powers of squarefree numbers.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reverse A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A363486 gives least prime index of greatest exponent.
A363487 gives greatest prime index of greatest exponent.
A364191 gives least prime index of least exponent.
A364192 gives greatest prime index of least exponent.
Cf. A000720, A000961, A051904, A061395, A071625, A130091, A289023, A367581, A367584, A367585, A367683.

mmk[q_]:=With[{mts=Length/@Split[q]},Sort[Table[Min@@Select[q,Count[q,#]==i&],{i,mts}]]];
Table[If[n==1,0,Max@@mmk[PrimePi/@Join@@ConstantArray@@@If[n==1,{},FactorInteger[n]]]],{n,1,100}]

a(n) = A061395(A367580(n)).
a(n^k) = a(n) for all positive integers n and k.
If n is a power of a squarefree number, a(n) = A055396(n).

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}]

A304491 Last or deepest exponent in the power-tower for n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 13 2018

nonn

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).

			We have 16 = 2^2^2, so a(16) = 2.
We have 64 = 2^6, so a(64) = 6.
We have 81 = 3^2^2, so a(81) = 2.
We have 256 = 2^2^3, so a(256) = 3.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304492, A304495.

a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},If[g==1,n,a[g]]]];
Array[a,100]

a(n)={my(t=n); while(t, n=t; t=ispower(t)); n} \\ Andrew Howroyd, Aug 26 2018

a(n) = A007916(A278028(n, A288636(n))).

A304495 Decapitate the power-tower for n, i.e., remove the last (deepest) exponent.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 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, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 13 2018

nonn

a(1) = 0 by convention.

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_1)^c(x_2)^c(x_3)^...^c(x_{k-1}).

			We have 64 = 2^6, so a(64) = 2.
We have 216 = 6^3, so a(216) = 6.
We have 256 = 2^2^3, so a(256) = 2^2 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A052409, A052410, A001597, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304491, A304492.

tow[n_]:=If[n==1,{},With[{g=GCD@@FactorInteger[n][[All,2]]},If[g===1,{n},Prepend[tow[g],n^(1/g)]]]];
Table[If[n==1,0,Power@@Most[tow[n]]],{n,100}]

A304495(n) = if(1==n,0,my(e, r, tow = List([])); while((e = ispower(n,,&r)) > 1, listput(tow, r); n = e;); n = 1; while(length(tow)>0, e = tow[#tow]; listpop(tow); n = e^n;); (n)); \\ Antti Karttunen, Jul 23 2018

a(m) <> 1 if m is a perfect power (A001597). - Michel Marcus, Jul 23 2018

Name edited and more terms from Antti Karttunen, Jul 23 2018

A304492 Position in the sequence of numbers that are not perfect powers (A007916) of the last or deepest exponent in the power-tower for n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 13 2018

nonn

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) = x_k.

Cf. A052409, A052410, A007916, A089723, A164337, A277562, A277564, A278028, A288636, A289023, A294336, A294337, A304481, A304491, A304495.

nn=100;
a[n_]:=If[n==1,1,With[{g=GCD@@FactorInteger[n][[All,2]]},If[g==1,n,a[g]]]];
rads=Union[Array[a,nn]];
Table[a[n],{n,nn}]/.Table[rads[[i]]->i,{i,Length[rads]}]

a(n) = A278028(n, A288636(n)).

A357310 a(n) is the number of j in the range 1 <= j <= n with the same maximal exponent in prime factorization as n.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 23 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000079 (positions of 1's), A051903, A058933, A289023.

f:= proc(n) option remember; `if`(n=1, 0,
      max(map(i-> i[2], ifactors(n)[2])))
    end:
b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+x^f(n)) end:
a:= n-> coeff(b(n), x, f(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 23 2022

Table[Length[Select[Range[n], If[# == 1, 0, Max @@ Last /@ FactorInteger[#]] == If[n == 1, 0, Max @@ Last /@ FactorInteger[n]] &]], {n, 1, 80}]
seq[max_] := Module[{e = Join[{0}, Table[Max @@ FactorInteger[n][[;; , 2]], {n, 2, max}]], c = Table[0, {max}]}, Do[c[[k]] = 1 + Count[e[[1 ;; k - 1]], e[[k]]], {k, 1, max}]; c]; seq[100] (* Amiram Eldar, Jan 05 2024 *)

lista(nmax) = {my(e = vector(nmax, k, if(k==1, 0, vecmax(factor(k)[,2]))), c); for(k = 1, nmax, c =  1; for(j = 1, k-1, c += (e[j] == e[k])); print1(c, ", "));} \\ Amiram Eldar, Jan 05 2024

a(n) = |{j <= n : A051903(j) = A051903(n)}|.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1/zeta(2)^2 + Sum_{k>=3} (1/zeta(k+1) - 1/zeta(k))^2 = 0.43029326822775728041... . - Amiram Eldar, Jan 05 2024

cp's OEIS Frontend

A072774 Powers of squarefree numbers.

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Formula

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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Haskell

Maple

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PARI

Python

Python

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A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

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Mathematica

Formula

A367583 Greatest element in row n of A367579 (multiset multiplicity kernel).

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Mathematica

Formula

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.

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Mathematica

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

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Maple

Mathematica

A304491 Last or deepest exponent in the power-tower for n.

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