A062759 -id:A062759 - cp's OEIS frontend

A059404 Numbers with different exponents in their prime factorizations.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200

Offset: 1

Labos Elemer, Jul 18 2001

Former name: Numbers k such that k/(largest power of squarefree kernel of k) is larger than 1.

Also numbers k = p(1)^alpha(1)* ... * p(r)^alpha(r) such that RootMeanSquare(alpha(1), ..., alpha(r)) is not an integer. - Ctibor O. Zizka, Sep 19 2008

			440 is in the sequence because 440 = 2^3*5*11 and it has two distinct exponents, 3 and 1.

Donald Alan Morrison, Table of n, a(n) for n = 1..10000
Donald Alan Morrison, Sage program

Complement of A072774 (powers of squarefree numbers).

Cf. A003557, A007947, A051904, A062759, A062760, A071625.

isA := n -> 1 < nops({seq(padic:-ordp(n, p), p in NumberTheory:-PrimeFactors(n))}): select(isA, [seq(1..190)]);  # Peter Luschny, Apr 14 2025

A059404Q[n_] := Length[Union[FactorInteger[n][[All, 2]]]] > 1;
Select[Range[200], A059404Q] (* Paolo Xausa, Jan 07 2025 *)

is(n)=#Set(factor(n)[,2])>1 \\ Charles R Greathouse IV, Sep 18 2015

from sympy import factorint
def ok(n): return len(set(factorint(n).values())) > 1
print([k for k in range(190) if ok(k)]) # Michael S. Branicky, Sep 01 2022

from math import isqrt
from sympy import mobius, integer_nthroot
def A059404(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def f(x): return n+1-(y:=x.bit_length())+sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

def isA(n): return 1 < len(set(valuation(n, p) for p in prime_divisors(n)))
print([n for n in range(1, 190) if isA(n)])  # Peter Luschny, Apr 14 2025

A062760(a(n)) > 1, i.e., a(n)/(A007947(a(n))^A051904(a(n))) = a(n)/A062759(a(n)) > 1.
A071625(a(n)) > 1. - Michael S. Branicky, Sep 01 2022
Sum_{n>=1} 1/a(n)^s = zeta(s) - 1 - Sum_{k>=1} (zeta(k*s)/zeta(2*k*s)-1) for s > 1. - Amiram Eldar, Mar 20 2025

A062770 n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Labos Elemer, Jul 18 2001

nonn

The sequence contains numbers m such that the exponents e are identical for all prime power factors p^e | m. It is clear from this alternate definition that m / K^E = 1 iff E is an integer. - Michael De Vlieger, Jun 24 2022

			Primes, squarefree numbers and perfect powers are here.
From _Michael De Vlieger_, Jun 24 2022 (Start):
144 cannot be in the sequence, since the exponents of its prime power factors differ. The squarefree kernel of 144 = 2^4 * 3^2 is 2*3 = 6. The largest power of 6 less than 144 is 36. 144/36 = 4, so it is not in the sequence.
216 is in the sequence because 216 = 2^3 * 3^3 is 2*3 = 6. But 216 = 6^3, hence 6^3 / 6^3 = 1. (End)

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

Cf. A062759, A062760, A007947, A003557, A051904, A005117, A001597, A072774.

Select[Range[2, 2^16], Length@ Union@ FactorInteger[#][[All, -1]] == 1 &] (* Michael De Vlieger, Jun 24 2022 *)

is(n)=ispower(n,,&n); issquarefree(n) && n>1 \\ Charles R Greathouse IV, Sep 18 2015

is(n)=#Set(factor(n)[,2])==1 \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius, integer_nthroot
def A062770(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

A062760(a(n)) = 1, i.e., a(n)/(A007947(a(n))^A051904(a(n))) = a(n)/A062759(a(n)) = 1.

a(n) = A072774(n+1). - Chai Wah Wu, Aug 19 2024

Offset corrected by Charles R Greathouse IV, Sep 18 2015

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

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

A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67

Offset: 1

Gus Wiseman, May 18 2018

nonn

Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.

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

Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.

acj[n_]:=Module[{f,m},f=FactorInteger[n];m=Max[Last/@f];Times@@Table[p[[1]]^(m-p[[2]]+1),{p,f}]];
Array[acj,100]

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[,2]) + 1); prod(i = 1, #f~, f[i,1]^e) / n);} \\ Amiram Eldar, Feb 12 2023

If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).

cp's OEIS Frontend

A059404 Numbers with different exponents in their prime factorizations.

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A062770 n/[largest power of squarefree kernel] equals 1; perfect powers of sqf-kernels (or sqf-numbers).

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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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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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A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

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Mathematica

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