A066636 -id:A066636 - cp's OEIS frontend

A373703 a(n) = A062760(n)*A066636(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 10, 1, 1, 1, 36, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 100, 1, 1, 1, 22, 15, 1, 1, 216, 1, 10, 1, 26, 1, 36, 1, 196, 1, 1, 1, 30, 1, 1, 21, 1, 1, 1, 1, 34, 1, 1, 1, 6, 1, 1, 15, 38, 1, 1, 1, 1000

Offset: 1

David James Sycamore, Jun 13 2024

nonn

If n has unequal prime exponents (a term in A059404), then a(n) > 1; otherwise a(n) = 1.

			a(12) = A062760(12)*A066636(12) = 2*3 = 6.
a(45) = A062760(45)*A066636(45) = 3*5 = 15.

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

Cf. A007947, A059404, A062760, A066636.

Table[Function[{r, s}, r^(Max[s] - Min[s])] @@ {Times @@ #[[All, 1]], #[[All, -1]]} &@ FactorInteger[n], {n, 120}] (* Michael De Vlieger, Jun 13 2024 *)

a(n) = if (n==1, 1, my(f=factor(n)); (factorback(f[, 1]))^(vecmax(f[,2])-vecmin(f[,2]))); \\ Michel Marcus, Jun 14 2024

For n > 1, a(n) = A007947(n)^k where k is the difference between the greatest and least exponents in the prime power factorization of n.

A066638 Smallest power of a squarefree number that is a multiple of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 13, 14, 15, 16, 17, 36, 19, 100, 21, 22, 23, 216, 25, 26, 27, 196, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 1000, 41, 42, 43, 484, 225, 46, 47, 1296, 49, 100, 51, 676, 53, 216, 55, 2744, 57, 58, 59

Offset: 1

Reinhard Zumkeller, Jan 09 2002

			a(40)=10^3 > 40 > 10=rad(40).

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

Cf. A005117, A066636.

Cf. A007947, A051903.

a066638 n = a007947 n ^ a051903 n  -- Reinhard Zumkeller, Jun 17 2015

rad[n_] := Times @@ First /@ FactorInteger[n]; mpe[n_] := Max @@ Last /@ FactorInteger[n]; mpe[1] = 0; a[n_] := rad[n]^mpe[n]; Table[a[n], {n, 1, 59}] (* Jean-François Alcover, Mar 27 2013 *)

a(n)=if(n==1,return(1)); my(f=factor(n));prod(i=1,#f~,f[i,1])^ vecmax(f[,2]) \\ Charles R Greathouse IV, Aug 21 2013

a(n) = rad(n)^mpe(n), (rad=A007947, mpe=A051903).

a(n) = A066636(n) * n. - Amiram Eldar, Sep 15 2023

A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 1, 1, 1, 1, 4

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) divides A003557 but is not equal to it.

a(n) is least d such that the prime power exponents of n/d are all equal; see also A066636. - David James Sycamore, Jun 13 2024

			n=1800: the squarefree kernel is 2*3*5 = 30 and 900 = 30^2 divides n, a(1800) = 2, the quotient of 1800/900.

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

Cf. A001694, A003557, A007947, A051904, A003557, A052485, A062759.
Cf. A059404 (n such that a(n)>1), A072774 (n such that a(n)=1).
Cf. A066636.

f:= proc(n) local F,m,t;
  F:= ifactors(n)[2];
  m:= min(seq(t[2],t=F));
  mul(t[1]^(t[2]-m),t=F)
end proc:
map(f, [$1..200]); # Robert Israel, Nov 03 2017

{1}~Join~Table[n/#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 104}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] -= e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A051904(n) = if(1==n,0,vecmin(factor(n)[, 2])); \\ After Charles R Greathouse IV's code
A062760(n) = n/(A007947(n)^A051904(n)); \\ Antti Karttunen, Sep 23 2017

a(n) = n/(A007947(n)^A051904(n)).

a(n) = n/A062759(n). - Amiram Eldar, Feb 12 2023

A304328 a(n) = n/(largest perfect power divisor of n).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 10 2018

nonn

Not all terms are squarefree numbers; for example, a(500) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001055, A001222, A001597, A001694, A005117, A007716, A007916, A052485, A052486, A059404, A066636, A091050, A160400, A203025, A294068, A303707, A304327, A304339.

Table[n/Last[Select[Divisors[n],#===1||GCD@@FactorInteger[#][[All,2]]>1&]],{n,100}]

a(n)={my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); n/m} \\ Andrew Howroyd, Aug 26 2018

a(n) * A203025(n) = n.

A304776 A weakening function. a(n) = n / A007947(n)^(A051904(n) - 1) where A007947 is squarefree kernel and A051904 is minimum prime exponent.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 12, 73, 74, 75, 76, 77, 78, 79, 80, 3, 82, 83

Offset: 1

Gus Wiseman, May 18 2018

nonn

This function takes powerful numbers (A001694) to weak numbers (A052485) and leaves weak numbers unchanged.

First differs from A052410 at a(72) = 12, A052410(72) = 72.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001597, A001694, A005117, A007947, A013929, A047966, A051903, A051904, A052485, A062759, A066636, A066638, A072774, A304768, A354090.

spr[n_]:=Module[{f,m},f=FactorInteger[n];m=Min[Last/@f];n/Times@@First/@f^(m-1)];
Array[spr,100]

A007947(n) = factorback(factorint(n)[, 1]);
A051904(n) = if((1==n),0,vecmin(factor(n)[, 2]));
A304776(n) = (n/(A007947(n)^(A051904(n)-1))); \\ Antti Karttunen, May 19 2022

a(n) = if(n == 1, 1, my(f = factor(n), p = f[, 1], e = f[, 2]); n / vecprod(p)^(vecmin(e) - 1)); \\ Amiram Eldar, Sep 12 2024

a(n) = n / A354090(n). - Antti Karttunen, May 19 2022

Sum_{k=1..n} a(k) ~ n^2 / 2. - Amiram Eldar, Sep 12 2024

Data section extended up to a(83) by Antti Karttunen, May 19 2022

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A373703 a(n) = A062760(n)*A066636(n).

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A066638 Smallest power of a squarefree number that is a multiple of n.

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A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

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A304328 a(n) = n/(largest perfect power divisor of n).

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A304776 A weakening function. a(n) = n / A007947(n)^(A051904(n) - 1) where A007947 is squarefree kernel and A051904 is minimum prime exponent.

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