A060476 -id:A060476 - cp's OEIS frontend

A055229 Greatest common divisor of largest square dividing n and squarefree part of n.

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

Offset: 1

Labos Elemer, Jun 21 2000

Record values occur at cubes of squarefree numbers: a(A062838(n)) = A005117(n) and a(m) < A005117(n) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000188, A007913, A007947, A008833.

Cf. A027748, A005117, A062838, A060476, A124010, A220218.

a055229 n = product $ zipWith (^) ps (map (flip mod 2) es) where
   (ps, es) = unzip $
              filter ((> 1) . snd) $ zip (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Oct 27 2015

a[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]}& /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 05 2014 *)

a(n)=my(c=core(n));gcd(c,n/c) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = gcd[A008833(n), A007913(n)].
Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p for odd e>1. - Vladeta Jovovic, Apr 30 2002
A220218(a(n)) = 1; A060476(a(n)) > 1 for n > 1. - Reinhard Zumkeller, Nov 30 2015
a(n) = core(n)*rad(n/core(n))/rad(n), where core = A007913 and rad = A007947. - Conjecture by Velin Yanev, proof by David J. Seal, Sep 19 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 + p^2 + p - 1)/(p^2 * (p + 1))) = 1.2249749939341923764... . - Amiram Eldar, Oct 08 2022

A368714 Numbers whose maximal exponent in their prime factorization is even.

Original entry on oeis.org

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208

Offset: 1

Amiram Eldar, Jan 04 2024

First differs from A240112 at n = 30.
Numbers k such that A051903(k) is even.
The asymptotic density of this sequence is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769... .

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

Cf. A051903, A240112.
Subsequences: A000290, A000302, A000583, A001014, A001016, A001025, A008454, A062503, A067259.
Similar sequences: A060476, A074661, A096432, A336064, A368715.

Select[Range[210], # == 1 || EvenQ[Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || !(vecmax(factor(k)[,2])%2), print1(k, ", ")));

A096432 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A004555.
Sequence is of positive density. - Charles R Greathouse IV, Dec 07 2012
The asymptotic density of this sequence is Sum_{p prime} (1/zeta(p) - 1/zeta(p-1)) = 0.8817562193... - Amiram Eldar, Oct 18 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..881 from R. J. Mathar)

Cf. A060476, A074661.

(Maple code for this entry and A074661)
M:=2000; ans1:=[]; ans2:=[];
for n from 1 to M do
t1:=op(2..-1, ifactors(n)); t2:=nops(t1);
m1:=0; for i from 1 to t2 do m1:=max(m1,t1[i][2]); od:
if isprime(1+m1) then ans1:=[op(ans1),n]; fi;
if isprime(m1) then ans2:=[op(ans2),n]; fi;
od:

Select[Range[2, 100], PrimeQ[1 + Max[FactorInteger[#][[;; , 2]]]] &] (* Amiram Eldar, Oct 18 2020 *)

isA096432(n) = if(n<2,0,isprime(vecmax(factor(n)[,2])+1))

A295661 Numbers with at least one odd exponent larger than one in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500, 504, 512, 513, 520, 536, 540

Offset: 1

Antti Karttunen, Nov 28 2017

nonn

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.1184861602... (= 1 - A065465). - Amiram Eldar, May 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of nonzero terms in A295662 and A295663.
Subsequence of A046099 (64 = 2^6, although a cube, is not in this sequence).
Differs from A060476 (256 = 2^8 is not a member of this sequence).
Complement of A335275.
Cf. A065465.

Select[Range[540], Count[FactorInteger[#][[All, -1]], ?(And[OddQ@ #, # > 1] &)] > 0 &] (* _Michael De Vlieger, Nov 28 2017 *)

from sympy import factorint
def ok(n):
    return max((e for e in factorint(n).values() if e%2), default=-1) > 1
print(list(filter(ok, range(541)))) # Michael S. Branicky, Aug 24 2021

A074661 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that max{e_2, e_3, ...} is prime.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 180, 184

Offset: 1

N. J. A. Sloane, Sep 18 2008

nonn

The old entry with this sequence number was a duplicate of A056594.
The largest exponent of the prime factors of n is prime. - Harvey P. Dale, Mar 09 2012
The asymptotic density of this sequence is Sum_{p prime} (1/zeta(p+1) - 1/zeta(p)) = 0.3391101054... - Amiram Eldar, Oct 18 2020

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

Cf. A096432, A060476.

Select[Range[200],PrimeQ[Max[Transpose[FactorInteger[#]][[2]]]]&] (* Harvey P. Dale, Mar 09 2012 *)

isA074661(n) = if(n<4,0,isprime(vecmax(factor(n)[,2])))

A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

Original entry on oeis.org

4, 12, 16, 18, 20, 24, 27, 28, 36, 44, 48, 50, 52, 54, 60, 64, 68, 72, 76, 80, 84, 90, 92, 98, 100, 108, 112, 116, 120, 124, 126, 132, 135, 140, 144, 148, 150, 156, 160, 162, 164, 168, 172, 176, 180, 188, 189, 192, 196, 198, 204, 208, 212, 216, 220, 228, 234, 236, 240, 242, 244

Offset: 1

Amiram Eldar, Jan 04 2024

Subsequence of A137257 and first differs from it at n = 51.
Numbers k such that gcd(k, A051903(k)) > 1.
Includes all the nonsquarefree terms of A336064.
The asymptotic density of this sequence is 1 - 1/zeta(2) - Sum_{k>=2} (1/(f(k+1, k) * zeta(k+1)) - 1/(f(k, k) * zeta(k))) = 0.24998449199080279703..., where f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).

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

Cf. A051903.
Subsequence of A013929 and A137257.
Similar sequences: A060476, A074661, A096432, A336064, A368714.

Select[Range[210], !CoprimeQ[#, Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 2, kmax, if(gcd(k, vecmax(factor(k)[,2])) > 1, print1(k, ", ")));

A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Sep 11 2024

Subsequence of A060476 and differs from it by not having the terms 1, 256, 768, 1280, 1792, 2304, ... .
Subsequence of A295661 and first differs from it at n = 51: A295661(51) = 432 is not a term of this sequence.
First differs from A325990 at n = 30: A325990(30) = 256 is not a term of this sequence.
Nonsquarefree numbers k such that A051903(k) is odd, or equivalently, numbers k such that A051903(k) is an odd number that is larger than 1.
The asymptotic density of this sequence is Sum_{k>=3} (-1)^(k+1) * (1 - 1/zeta(k)) = 0.11615617754774636364... .

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

Complement of A368714 within A013929.

Cf. A051903, A060476, A295661, A325990.

q[n_] := n > 1 && OddQ[n]; Select[Range[500], q[Max[FactorInteger[#][[;; , 2]]]] &]

is(k) = k > 1 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));

A325990 Numbers with more than one perfect factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A060476 in lacking 1 and having 432.

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Positions of terms > 1 in A325989.

Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A295661, A325990 and A376142 at n = 24: A295661(24) = A325990(24) = A376142(24) = 216 = 2^3 * 3^3 is not a term of this sequence.
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... .
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .

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

Subsequence of A295661.
Subsequences: A065036, A143610, A163569.
Cf. A060476, A065465, A325990, A376142, A377845.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]

is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;

cp's OEIS Frontend

A055229 Greatest common divisor of largest square dividing n and squarefree part of n.

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A368714 Numbers whose maximal exponent in their prime factorization is even.

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A096432 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that 1 + max{e_2, e_3, ...} is a prime.

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A295661 Numbers with at least one odd exponent larger than one in their prime factorization.

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A074661 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n; sequence gives n such that max{e_2, e_3, ...} is prime.

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A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

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A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

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A325990 Numbers with more than one perfect factorization.

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