A327501 -id:A327501 - cp's OEIS frontend

A327502 a(n) = n/A327501(n), where A327501(n) is the maximum divisor of n that is 1 or not a perfect power.

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 9, 1, 1, 1, 1, 16, 1, 1, 1, 2, 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, 32, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 16 2019

This maximum divisor is given by A327501.

A multiset is aperiodic if its multiplicities are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Heinz numbers of aperiodic multisets are numbers that are not perfect powers (A007916).

			The divisors of 36 that are 1 or not a perfect power are {1, 2, 3, 6, 12, 18}, so a(36) = 36/18 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A000961, A001597, A007916, A303386, A327501.

Table[n/Max[Select[Divisors[n],GCD@@Last/@FactorInteger[#]==1&]],{n,100}]

A327502(n) = if(n==1, 1, n/vecmax(select(x->((x>1) && !ispower(x)), divisors(n)))); \\ Antti Karttunen, Sep 19 2019 (after program given by Michel Marcus for A327501)

a(n) = n/A327501(n).

A327503 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).

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

Gus Wiseman, Sep 16 2019

nonn

First differs from A052409 and A158378 at a(216) = 2, A052409(216) = A158378(216) = 3.
A multiset is aperiodic if its multiplicities are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Heinz numbers of aperiodic multisets are numbers that are not perfect powers (A007916).
a(n) does not depend only on the prime signature of n. A351948 gives the positions where a(A046523(n)) <> a(n). n = 125000 is the first time this happens, see the examples. - Antti Karttunen, Apr 03 2022

			The transformation A327502 takes 144 -> 2 -> 1, so a(144) = 2.
From _Antti Karttunen_, Apr 03 2022: (Start)
For n = 1728 = 2^6 * 3^3, A327501(1728) = 864 = 2^5 * 3^3, and therefore A327502(1728) = 1728/864 = 2. A327501(2) = 2, thus A327502(2) = 2/2 = 1, so we reached 1 (= A327502(1)) in two steps, and therefore a(1728) = 2.
For n = 125000 = 2^3 * 5^6, A327501(125000) = 31250 = 2^1 * 5^6, and therefore A327502(125000) = 125000/31250 = 4. A327501(4) = 2, thus A327502(4) = 4/2 = 2, from which we reach 1 in one more step, therefore a(125000) = 3.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.
Cf. A000961, A001597, A007916 (positions of 1's), A052409, A158378, A303386, A327501, A327502, A351948.
Cf. also A327500.

Table[Length[FixedPointList[#/Max[Select[Divisors[#],GCD@@Last/@FactorInteger[#]==1&]]&,n]]-2,{n,100}]

A327502(n) = if(n==1, 1, n/vecmax(select(x->((x>1) && !ispower(x)), divisors(n))));
A327503(n) = { my(u=A327502(n)); if(u==n, 0, 1+A327503(u)); }; \\ Antti Karttunen, Apr 02 2022

a(2^n) = n.

Data section extended up to 105 terms by Antti Karttunen, Apr 02 2022

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

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

cp's OEIS Frontend

A327502 a(n) = n/A327501(n), where A327501(n) is the maximum divisor of n that is 1 or not a perfect power.

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A327503 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum divisor that is 1 or not a perfect power (A327501, A327502).

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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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