A072414 -id:A072414 - 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

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 3

Offset: 1

Labos Elemer, Jun 17 2002

The sums of the first 10^k terms, for k = 1, 2, ..., are 14, 168, 1779, 17959, 180665, 1808044, 18084622, 180856637, 1808585068, 18085891506, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.8085... . - Amiram Eldar, Sep 10 2022

			n = 288 = 2*2*2*2*2*3*3; lcm(5,2) = 10; Product(5,2) = 10, max(5,2) = 5;
n = 180 = 2*2*3*3*5; lcm(2,2,1) = 2; Product(2,2,1) = 4; max(2,2,1) = 2; it deviates both from maximum of exponents (A051903, for the first time at n=72), and product of exponents (A005361, for the first time at n=36).
For n = 36 = 2*2*3*3 = 2^2 * 3^2 we have a(36) = lcm(2,2) = 2.
For n = 72 = 2*2*2*3*3 = 2^3 * 3^2 we have a(72) = lcm(2,3) = 6.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(2,4) = 4.
For n = 360 = 2^3 * 3^2 * 5^1 we have a(360) = lcm(1,2,3) = 6.

Cf. A028234, A067029, A072412-A072414, A273058, A284569, A290103.
Similar sequences: A001222 (sum of exponents), A005361 (product), A051903 (maximal exponent), A051904 (minimal exponent), A052409 (gcd of exponents), A267115 (bitwise-and), A267116 (bitwise-or), A268387 (bitwise-xor).
Cf. also A055092, A060131.
Differs from A290107 for the first time at n=144.
After the initial term, differs from A157754 for the first time at n=360.

Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n) = lcm(factor(n)[,2]); \\ Michel Marcus, Mar 25 2017

from sympy import lcm, factorint
def a(n):
    l=[]
    f=factorint(n)
    for i in f: l+=[f[i],]
    return lcm(l)
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 25 2017

a(1) = 1; for n > 1, a(n) = lcm(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 09 2016
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A284569(A156552(n)).
a(n) = A290103(A181819(n)).
a(A289625(n)) = A002322(n).
a(A290095(n)) = A055092(n).
a(A275725(n)) = A060131(n).
a(A260443(n)) = A277326(n).
a(A283477(n)) = A284002(n). (End)

a(1) = 1 prepended and the data section filled up to 120 terms by Antti Karttunen, Aug 09 2016

A072413 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the product of the exponents.

Original entry on oeis.org

36, 100, 144, 180, 196, 216, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1000, 1008, 1044, 1080, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1512, 1521

Offset: 1

Labos Elemer, Jun 17 2002

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 2, 29, 348, 3548, 35761, 358258, 3583892, 35843109, 358440763, ... . Apparently, the asymptotic density of this sequence exists and equals 0.03584... . - Amiram Eldar, Sep 09 2022

			k = 36 = 2*2*3*3; exponent set = {2,2}; LCM = 2, product = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005361, A051903, A051904, A052409, A052486, A072411-A072414.

Select[Range@ 1600, LCM @@ # != Times @@ # &@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, May 15 2016 *)

is(n)=my(f=factor(n)[,2]); n>9 && lcm(f)!=factorback(f) \\ Charles R Greathouse IV, Jan 14 2017

A005361(a(n)) != A072411(a(n)).

A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

Original entry on oeis.org

72, 108, 200, 288, 360, 392, 432, 500, 504, 540, 600, 648, 675, 756, 792, 800, 864, 936, 968, 972, 1125, 1152, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1440, 1500, 1568, 1656, 1800, 1836, 1944, 1960, 2000, 2016, 2052, 2088, 2160

Offset: 1

Labos Elemer, Jun 17 2002

nonn

This sequence differs from the Achilles numbers (A052486).

			k = 360 = 2*2*2*3*3*5, exponent set = {3,2,1}; LCM=6, max=3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A051903, A051904, A052409, A052486, A072411, A072413, A072414.

Select[Range[2000], LCM @@ (e = FactorInteger[#][[;; , 2]]) != Max[e] &] (* Amiram Eldar, Jul 30 2022 *)

is(n)=my(f=factor(n)[,2]); n>9 && vecmax(f)!=lcm(f) \\ Charles R Greathouse IV, Oct 16 2015

A051903(a(n)) != A072411(a(n)).

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Scheme

Formula

Extensions

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A072413 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the product of the exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula