A052409 -id:A052409 - cp's OEIS frontend

A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410

Offset: 0

Alford Arnold, Aug 27 2000

Note that for n>0 the prime divisors of a(n) are consecutive primes starting with 2. All of the least prime signatures (A025487) are included; with the other values forming A056808.
Using the formula, terms of b(n)= a(n)/A057334(n) are: 1, 1, 2, 2, 4, 4, 6, 6, 8, ..., indeed a(n) repeated. - Michel Marcus, Feb 09 2014
a(n) is the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic). This composition (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. A number is normal if its prime indices cover an initial interval of positive integers. Unsorted prime signature is the sequence of exponents in a number's prime factorization. - Gus Wiseman, Apr 19 2020

			From _Gus Wiseman_, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     12: {1,1,2}
     18: {1,2,2}
     30: {1,2,3}
     16: {1,1,1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     60: {1,1,2,3}
     54: {1,2,2,2}
     90: {1,2,2,3}
    150: {1,2,3,3}
    210: {1,2,3,4}
     32: {1,1,1,1,1}
     48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000120, A057334, A055932 and A056808.
Cf. A324939.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
The reversed version is A334031.
A partial inverse is A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A005867, A029931, A048793, A052409, A056239, A066099, A112798, A124767, A228351, A233249, A333220, A345974.
Related to A019565 via A122111 and to A000079 via A336321.

Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)

mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); v;} \\ Michel Marcus, Feb 09 2014

A057335(n) = if(0==n,1,prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020

a(n) = A057334(n) * a (repeated).
A334032(a(n)) = n; a(A334032(n)) = A071364(n). - Gus Wiseman, Apr 19 2020
a(n) = A122111(A019565(n)); A019565(n) = A122111(a(n)). - Peter Munn, Jul 18 2020
a(n) = A336321(2^n). - Peter Munn, Mar 04 2022
Sum_{n>=0} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

New primary name from Antti Karttunen, Jul 20 2020

A333632 Rotational period of the k-th composition in standard order; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 1, 2, 2, 3, 1, 3, 3, 4, 2, 3, 1, 4, 3, 2, 4, 5, 2, 3, 3, 4, 3, 4, 2, 5, 3, 4, 4, 5, 4, 5, 5, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4

Offset: 0

Gus Wiseman, Apr 12 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The a(299) = 5 rotations:
  (1,1,3,2,2)
  (1,3,2,2,1)
  (3,2,2,1,1)
  (2,2,1,1,3)
  (2,1,1,3,2)
The a(9933) = 4 rotations:
  (1,2,1,3,1,2,1,3)
  (1,3,1,2,1,3,1,2)
  (2,1,3,1,2,1,3,1)
  (3,1,2,1,3,1,2,1)

Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
The version for binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
Compositions by number of distinct rotations are A333941.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Equal runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632 (this sequence).
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[Array[RotateRight[stc[n],#]&,DigitCount[n,2,1]]]],{n,0,100}]

a(n) = A000120(n)/A138904(n) = A302291(n) - A023416(n)/A138904(n).

A075802 Characteristic function of perfect powers, A001597.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 13 2002

nonn

Not multiplicative: for example, a(8)=a(9)=1, but a(72)=0. - Franklin T. Adams-Watters, Sep 09 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Index entries for characteristic functions

Cf. A001597, A007916, A052409, A057427, A072292, A112526.

a075802 1 = 1
a075802 n = signum $ a052409 n - 1  -- Reinhard Zumkeller, May 26 2012

a[n_] := Boole[GCD @@ FactorInteger[n][[All, 2]] > 1]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 12 2011 *)

from sympy import perfect_power
def A075802(n): return int(bool(perfect_power(n))) if n>1 else 1 # Chai Wah Wu, Mar 11 2025

a(n) = A057427(A052409(n) - 1);

a(A001597(n))=1 and a(A007916(n))=0.

A366275 The Cat's tongue permutation: a(n) = A163511(A057889(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 15, 12, 25, 10, 7, 32, 81, 54, 45, 36, 75, 30, 21, 24, 125, 50, 35, 20, 49, 14, 11, 64, 243, 162, 135, 108, 225, 90, 63, 72, 375, 150, 105, 60, 147, 42, 33, 48, 625, 250, 175, 100, 245, 70, 55, 40, 343, 98, 77, 28, 121, 22, 13, 128, 729, 486, 405, 324, 675, 270, 189, 216, 1125

Offset: 0

Antti Karttunen, Oct 06 2023

"Cat's tongue" refers to the look of the scatter plot of this sequence.

Cf. A000040, A000225, A007814, A057889, A163511, A209229, A290251, A366276 (inverse map), A366277 (fixed points of map n -> a(n)), A366278, A366279, A366280, A366281 [= A052409(a(n))], A366282 [= a(n)-n], A366283 [= gcd(n,a(n))].

Cf. also A163511, A253563, A366263 (compare the scatter plots).

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));

from sympy import prime
def A366275(n):
    if n:
        k, c, m = int(bin(n>>(r:=(~n & n-1).bit_length()))[:1:-1],2)<>= s+1
        return m*prime(c)
    return 1 # Chai Wah Wu, Oct 08 2023

For n >= 0, A001222(a(n)) = A290251(n).
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [Like A163511, also this permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, a(2*n) = 2*a(n).
For n >= 1, a(A000225(n)) = A000040(n).

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

A138904 Number of rotational symmetries in the binary expansion of a number.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Max Sills, Apr 03 2008, Apr 04 2008

Mersenne numbers of form (2^n - 1) have n rotational symmetries.
For prime length binary expansions these are the only nontrivial symmetries.
For composite length expansions it seems that when the number of symmetries is nontrivial it is equal to a factor of the length. We're working on an explicit formula.
Discovered in the context of random circulant matrices, examining if there's a correlation between degrees of freedom and number of symmetries in the first row.
When combined with A138954, these two sequences should give a full account of the number of redundant rows in a circulant square matrix with at most two distinct values, where a(n) is the encoding of the first row of the matrix into binary such that value a = 1 and value b = 0.
Discovered on the night of Apr 02, 2008 by Maxwell Sills and Gary Doran.
Conjecture: For binary expansions of length n, there are d(n) distinct values that will show up as symmetries, where d is the divisor function. The symmetry values will be precisely the divisors of n.
Example: for binary expansions of length 12, one sees that d(12) = 6 distinct values show up as symmetries (1, 2, 3, 4, 6, 12).
Conjecture: For numbers whose binary expansion has length n which has proper divisors which are all coprime: There will be only one number of length n with n symmetries. That number is 2^n - 1. For each proper divisor d (excluding 1), you can generate all numbers of length n that have n/d symmetries like so: (2^0 + 2^d + 2^2d ... 2^(n-d)) * a, where 2^(d-1) <= a < (2^d) - 1. The rest of the expansions of length n will have only the trivial symmetry.
Also the number of rotational symmetries of the n-th composition in standard order (graded reverse-lexicographic). This composition (row n of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. - Gus Wiseman, Apr 19 2020
From Gus Wiseman, Apr 19 2020: (Start)
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Compositions by sum and number of distinct rotations are A333941.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.
(End).

			a(10) = 2 because the binary expansion of 10 is 1010 and it has two rotational symmetries (including identity).

Maxwell Sills and Gary Doran, Table of n, a(n) for n = 0..99

Cf. A136441, A138954.

Table[IntegerLength[n,2]/Length[Union[Array[RotateRight[IntegerDigits[n,2],#]&,IntegerLength[n,2]]]],{n,100}] (* Gus Wiseman, Apr 19 2020 *)

a(n) = A070939(n)/A302291(n) = A000120(n)/A333632(n). - Gus Wiseman, Apr 19 2020

A258409 Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 2, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 2, 1, 78, 1

Offset: 2

Ivan Neretin, May 29 2015

nonn

a(n) = 1 for even n; a(p) = p-1 for prime p.
a(n) is even for odd n (since all divisors of n are odd).
It appears that a(n) = A052409(A005179(n)), i.e., it is the largest integer power of the smallest number with exactly n divisors. - Michel Marcus, Nov 10 2015
Conjecture: GCD of all (p-1) for prime p|n. - Thomas Ordowski, Sep 14 2016
Conjecture is true, because the set of numbers == 1 (mod g) is closed under multiplication. - Robert Israel, Sep 14 2016
Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164. - Gus Wiseman, Oct 16 2019

			65 has divisors 1, 5, 13, and 65, hence a(65) = gcd(1-1,5-1,13-1,65-1) = gcd(0,4,12,64) = 4.

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A049559, A057237, A060680, A063994, A187730, A027750.
Cf. A084190 (similar but with LCM).
Looking at prime indices instead of divisors gives A328167.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000005, A060681, A060683, A193829, A289508.

a258409 n = foldl1 gcd $ map (subtract 1) $ tail $ a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015

f:= n -> igcd(op(map(`-`,numtheory:-factorset(n),-1))):
map(f, [$2..100]); # Robert Israel, Sep 14 2016

Table[GCD @@ (Divisors[n] - 1), {n, 2, 100}]

a(n) = my(g=0); fordiv(n, d, g = gcd(g, d-1)); g; \\ Michel Marcus, May 29 2015

a(n) = gcd(apply(x->x-1, divisors(n))); \\ Michel Marcus, Nov 10 2015

a(n)=if(n%2==0, return(1)); if(n%3==0, return(2)); if(n%5==0 && n%4 != 1, return(2)); gcd(apply(p->p-1, factor(n)[,1])) \\ Charles R Greathouse IV, Sep 19 2016

A332214 Mersenne-prime fixing variant of permutation A163511: a(n) = A332212(A163511(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 6, 7, 16, 27, 18, 49, 12, 21, 14, 5, 32, 81, 54, 343, 36, 147, 98, 25, 24, 63, 42, 35, 28, 15, 10, 31, 64, 243, 162, 2401, 108, 1029, 686, 125, 72, 441, 294, 175, 196, 75, 50, 961, 48, 189, 126, 245, 84, 105, 70, 155, 56, 45, 30, 217, 20, 93, 62, 11, 128, 729, 486, 16807, 324, 7203, 4802, 625, 216, 3087, 2058, 875

Offset: 0

Antti Karttunen, Feb 09 2020

Any Mersenne prime (A000668) times any power of 2, i.e., sequence A335431, is fixed by this map (note the indexing), including also all even perfect numbers. It is not currently known whether there are any additional fixed points.

Because a(n) has the same prime signature as A163511(n), it implies that applying A046523 and A052409 to this sequence gives the same results as with A163511, namely, sequences A278531 and A365805. - Antti Karttunen, Oct 09 2023

Cf. A163511, A332211, A332212, A332215 (inverse permutation).
Cf. A278531 [= A046523(a(n))], A290251 [= A001222(a(n))], A365805 [= A052409(a(n))], A366372 [= a(n)-n], A366373 [= gcd(n,a(n))], A366374 (numerator of n/a(n)), A366375 (denominator of n/a(n)), A366376.
Cf. A000043, A000668, A000396, A324200, A335431 (conjectured to give all the fixed points).

PARI
```
A332214(n) = A332212(A163511(n));
```

\\ Needs precomputed data for A332211:
v332211 = readvec("b332211_to.txt"); \\ Prepared with gawk ' { print $2 } ' < b332211.txt > b332211_to.txt
A332211(n) = v332211[n];
A332214(n) = if(!n, 1, my(i=1, p=A332211(i), t=1); while(n>1, if(!(n%2), (t*=p), i++; p=A332211(i)); n >>= 1); (t*p)); \\ Antti Karttunen, Oct 09 2023

a(n) = A332212(A163511(n)).

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).

A025479 Largest exponents of perfect powers (A001597).

Original entry on oeis.org

2, 2, 3, 2, 4, 2, 3, 5, 2, 2, 6, 4, 2, 2, 3, 7, 2, 2, 2, 3, 2, 5, 8, 2, 2, 3, 2, 2, 2, 2, 9, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 10, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 11, 2, 7, 3, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 12, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 8, 2, 3, 2, 2, 2

Offset: 1

David W. Wilson

Greatest common divisor of all prime-exponents in canonical factorization of n-th perfect power. - Reinhard Zumkeller, Oct 13 2002

Asymptotically, 100% of the terms are 2, since the density of cubes and higher powers among the squares and higher powers is 0. - Daniel Forgues, Jul 22 2014

Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A001597, A025478, A052409, A124010, A322969.

a025479 n = a025479_list !! (n-1)  -- a025479_list is defined in A001597.
-- Reinhard Zumkeller, Mar 28 2014, Jul 15 2012

N:= 10^6: # to get terms corresponding to all perfect powers <= N
V:= Vector(N,storage=sparse);
V[1]:= 2:
for p from 2 to ilog2(N) do
  V[[seq(i^p,i=2..floor(N^(1/p)))]]:= p
od:
r,c,A := ArrayTools:-SearchArray(V):
convert(A,list); # Robert Israel, Apr 25 2017

Prepend[DeleteCases[#, 0], 2] &@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, e, 0] &@ FactorInteger@ n, {n, 10^4}] (* Michael De Vlieger, Apr 25 2017 *)

print1(2,", "); for(k=2, 3^8, if(j=ispower(k),print1(j,", "))) \\ Hugo Pfoertner, Jan 01 2019

a(n) = A052409(A001597(n)). - Reinhard Zumkeller, Oct 13 2002

A001597(n) = A025478(n)^a(n). - Reinhard Zumkeller, Mar 28 2014

Definition corrected by Daniel Forgues, Mar 07 2009

cp's OEIS Frontend

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