A288636 -id:A288636 - cp's OEIS frontend

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

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

A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

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

Gus Wiseman, Oct 28 2017

nonn

Möbius-transform of A294337. - Antti Karttunen, Jun 12 2018

			The a(4096) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

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

Cf. A000041, A001055, A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294337, A294338, A294339, A375598, A375599.

Array[1+Sum[#0[g],{g,Rest[Divisors[GCD@@FactorInteger[#1][[All,2]]]]}]&,200]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d))); \\ Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} a(d). - Antti Karttunen, Jun 12 2018, after Mathematica-code.

a(n) = A294337(A052409(n)) for n >= 2. - Pontus von Brömssen, Aug 20 2024

More terms from Antti Karttunen, Jun 12 2018

A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 6, 13, 4, 9, 2, 17, 6, 19, 10, 9, 4, 23, 6, 5, 4, 3, 14, 29, 8, 31, 2, 9, 4, 25, 4, 37, 4, 9, 10, 41, 8, 43, 22, 15, 4, 47, 6, 7, 10, 9, 26, 53, 6, 25, 14, 9, 4, 59, 18, 61, 4, 21, 2, 25, 8, 67, 34, 9, 8, 71, 6, 73, 4, 15, 38

Offset: 1

Gus Wiseman, Nov 26 2023

nonn

As an operation on multisets, this is represented by A367579.

			90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so a(90) = 12.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Positions of prime(k) are prime powers prime(k)^i, rows of A051128.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
Agrees with A181819 at positions A367683, counted by A367682.
Rows of A367579 have this rank, sum A367581, max A367583, min A055396.
Positions of first appearances are A367584, sorted A367585.
Positions of powers of 2 are A367586.
Divides n at positions A367685, counted by A367684.
The opposite version (cokernel) is A367859.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A001597, A005117, A020639, A051904, A052409, A062770, A175781, A238747, A288636, A353742, A367582.

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

a(n^k) = a(n) for all positive integers n and k.
A001221(a(n)) = A071625(n).
A001222(a(n)) = A001221(n).
If n is squarefree, a(n) = A020639(n)^A001222(n).
A056239(a(n)) = A367581(n).

A278028 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 x_1, ..., x_k.

Original entry on oeis.org

1, 2, 1, 1, 3, 4, 5, 1, 2, 2, 1, 6, 7, 8, 9, 10, 11, 1, 1, 1, 12, 13, 14, 15, 16, 17, 18, 19, 3, 1, 20, 2, 2, 21, 22, 23, 24, 1, 3, 25, 26, 27, 4, 1, 28, 29, 30, 31

Offset: 1

N. J. A. Sloane, Nov 09 2016

Row lengths are A288636(n). - Gus Wiseman, Jun 12 2017

			Rows 2 through 32 are:
1,
2,
1, 1,
3,
4,
5,
1, 2,
2, 1,
6,
7,
8,
9,
10,
11,
1, 1, 1,
12,
13,
14,
15,
16,
17,
18,
19,
3, 1,
20,
2, 2,
21,
22,
23,
24,
1, 3,
...

N. J. A. Sloane, Table of n, a(n) for n = 1..20181
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

See A277564 for another version.

Cf. A007916, A089723, A277562, A288636.

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.

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).

A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 7, 2, 4, 4, 10, 2, 7, 2, 7, 4, 4, 2, 10, 4, 4, 6, 7, 2, 8, 2, 12, 4, 4, 4, 12, 2, 4, 4, 10, 2, 8, 2, 7, 7, 4, 2, 15, 4, 7, 4, 7, 2, 10, 4, 10, 4, 4, 2, 13, 2, 4, 7, 16, 4, 8, 2, 7, 4, 8, 2, 16, 2, 4, 7, 7, 4, 8, 2, 15, 10, 4, 2, 13, 4, 4, 4, 10, 2, 13, 4, 7, 4, 4, 4, 18, 2, 7, 7, 12, 2, 8, 2, 10, 8

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(12) = 7 ways are: 2^12, 4^6, 8^4, 8^(2^2), 16^3, 64^2, 4096.

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

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294338, A294339.

a[n_]:=1+Sum[a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Table[a[2^n],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A294336(n) = if(1==n,n,sumdiv(A052409(n),d,A294336(d)));
A294337(n) = sumdiv(n,d,A294336(d));
\\ Or alternatively, after Mathematica-code as:
A294337(n) = A294336(2^n); \\ Antti Karttunen, Jun 12 2018

a(n) = Sum_{d|n} A294336(d) = A294336(A000079(n)). - Antti Karttunen, Jun 12 2018

More terms from Antti Karttunen, Jun 12 2018

A294338 Number of ways to write n as a finite power-tower of positive integers greater than one, allowing both left and right nesting of parentheses.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 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, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 28 2017

nonn

			The a(16) = 5 ways are: 16, 4^2, (2^2)^2, 2^4, 2^(2^2).

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A001597, A007916, A052409, A052410, A089723, A164336, A277562, A284639, A288636, A294336, A294337, A294339.

A294338 := proc(n)
    local expo,g,a,d ;
    if n =1 then
        return 1;
    end if;
    # compute gcd of the set of prime power exponents (A052409)
    ifactors(n)[2] ;
    [ seq(op(2,ep),ep=%)] ;
    igcd(op(%)) ;
    # set of divisors of A052409 (without the 1)
    g := numtheory[divisors](%) minus {1} ;
    a := 0 ;
    for d in g do
        # recursive (sort of convolution) call
        a := a+ procname(d)*procname(root[d](n)) ;
    end do:
    1+a ;
end proc:
seq(A294338(n),n=1..120) ; # R. J. Mathar, Nov 27 2017

a[n_]:=1+Sum[a[n^(1/g)]*a[g],{g,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}];
Array[a,100]

A381212 a(n) is the smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Feb 19 2025

The corresponding largest elements are given by A081812.
The positions of terms > 1 are given by A001694.
Records of a(n) = 2, 3, 4, 5,.. appear at n=4=2^2, 27=3^3, 625=5^4, 3125=5^5, 117649=7^6, 823543=7^7 ,... (subsequence A051647).- R. J. Mathar, Mar 05 2025

			a(36) = 2 because 36 = 2^2*3^2, the set of these bases and exponents is {2, 3} and its smallest element is 2.
a(31500) = 1 because 31500 = 2^2*3^2*5^3*7^1, the set of these bases and exponents is {1, 2, 3, 5, 7} and its smallest element is 1.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A001694, A081812, A288636, A331048, A381201, A381202, A381203, A381204, A381205, A381214.

A381212 := proc(n)
    local a,pe;
    a := n ;
    for pe in ifactors(n)[2] do
        a := min(a,op(1,pe),op(2,pe)) ;
    end do:
    a ;
end proc:
seq(A381212(n),n=2..100) ; # R. J. Mathar, Mar 05 2025

A381212[n_] := Min[Flatten[FactorInteger[n]]];
Array[A381212, 100, 2]

a(n) = my(f=factor(n)); vecmin(setunion(Set(f[,1]), Set(f[,2]))); \\ Michel Marcus, Feb 20 2025

cp's OEIS Frontend

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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A294336 Number of ways to write n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.

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A367580 Multiset multiplicity kernel (MMK) of n. Product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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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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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).

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A294337 Number of ways to write 2^n as a finite power-tower a^(b^(c^...)) of positive integers greater than one.