A089384 -id:A089384

Search: a089384 -id:a089384

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page 1

A366521

Largest squarefree divisor of n which is <= sqrt(n).

+10
2

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 5, 1, 2, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 2, 5, 2, 1, 6, 7, 5, 3, 2, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 2, 5, 6, 1, 2, 3, 7, 1, 6, 1, 2, 5, 2, 7, 6, 1, 5, 3, 2, 1, 7, 5, 2, 3, 2, 1, 6, 7, 2, 3, 2, 5, 6, 1, 7, 3, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

MATHEMATICA

Table[Last[Select[Divisors[n], # <= Sqrt[n] && SquareFreeQ[#] &]], {n, 100}]

PROG

(PARI) a(n) = {my(m=1); fordiv(n, d, if(d^2 <= n && issquarefree(d), m=max(m, d))); m} \\ Andrew Howroyd, Oct 11 2023

CROSSREFS

Cf. A007947, A033676, A089384, A366522.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Oct 11 2023

STATUS

approved

A366522

Largest squarefree divisor of n which is < sqrt(n), for n >= 2; a(1) = 1.

+10
2

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 5, 1, 2, 3, 2, 5, 3, 1, 2, 3, 5, 1, 6, 1, 2, 5, 2, 1, 6, 1, 5, 3, 2, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 2, 5, 6, 1, 2, 3, 7, 1, 6, 1, 2, 5, 2, 7, 6, 1, 5, 3, 2, 1, 7, 5, 2, 3, 2, 1, 6, 7, 2, 3, 2, 5, 6, 1, 7, 3, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

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

MATHEMATICA

Join[{1}, Table[Last[Select[Divisors[n], # < Sqrt[n] && SquareFreeQ[#] &]], {n, 2, 100}]]

PROG

(PARI) a(n) = {my(m=1); fordiv(n, d, if(d^2 < n && issquarefree(d), m=max(m, d))); m} \\ Andrew Howroyd, Oct 11 2023

CROSSREFS

Cf. A007947, A060775, A089384, A366521.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Oct 11 2023

STATUS

approved

A365837

Largest proper square divisor of n, for n >= 2; a(1) = 1.

+10
1

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 1, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 16, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 1, 49, 9, 25

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

f:= proc(n) local F, t;

if issqr(n) then

n/min(numtheory:-factorset(n))^2

else

F:= ifactors(n)[2];

mul(t[1]^(2*floor(t[2]/2)), t=F)

end proc:

f(1):= 1:

map(f, [$1..100]); # Robert Israel, Nov 20 2023

MATHEMATICA

Join[{1}, Table[Last[Select[Divisors[n], # < n && IntegerQ[Sqrt[#]] &]], {n, 2, 100}]]

f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f @@@ fct/If[AllTrue[fct[[;; , 2]], EvenQ], fct[[1, 1]]^2, 1]]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

PROG

(PARI) a(n) = if (n==1, 1, my(d=divisors(n)); vecmax(select(issquare, Vec(d, #d-1)))); \\ Michel Marcus, Oct 17 2023

(Python)

from math import prod

from sympy import factorint

def A365837(n):

if n<=1: return 1

f = factorint(n)

return prod(p**(e&-2) for p, e in f.items())//(min(f)**2 if all(e&1^1 for e in f.values()) else 1) # Chai Wah Wu, Oct 20 2023

CROSSREFS

Cf. A008833, A032742, A063928, A085392, A089384, A093803, A366524.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Oct 17 2023

STATUS

approved

A366649

Largest prime power (including 1) proper divisor of n, for n >= 2; a(1) = 1.

+10
1

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 4, 1, 7, 5, 8, 1, 9, 1, 5, 7, 11, 1, 8, 5, 13, 9, 7, 1, 5, 1, 16, 11, 17, 7, 9, 1, 19, 13, 8, 1, 7, 1, 11, 9, 23, 1, 16, 7, 25, 17, 13, 1, 27, 11, 8, 19, 29, 1, 5, 1, 31, 9, 32, 13, 11, 1, 17, 23, 7, 1, 9, 1, 37, 25, 19, 11, 13, 1, 16, 27, 41, 1, 7, 17

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

f:= proc(n) local F, t;

F:= ifactors(n)[2];

if nops(F) = 1 then n/F[1, 1]

else max(map(t -> t[1]^t[2], F))

end proc:

f(1):= 1:

map(f, [$1..100]); # Robert Israel, Nov 19 2023

MATHEMATICA

Join[{1}, Table[Last[Select[Divisors[n], # < n && (# == 1 || PrimePowerQ[#]) &]], {n, 2, 85}]]

a[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 1]]^(f[[1, 2]] - 1), Max[Power @@@ f]]]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

PROG

(PARI) a(n) = if (n==1, 1, my(d=divisors(n)); vecmax(select(x->(isprimepower(x) || (x==1)), Vec(d, #d-1)))); \\ Michel Marcus, Oct 17 2023

CROSSREFS

Cf. A032742, A034699, A063928, A085392, A089384, A093803.

KEYWORD

nonn,look

AUTHOR

Ilya Gutkovskiy, Oct 17 2023

STATUS

approved

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