A062760 -id:A062760 - 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.

A059404 Numbers with different exponents in their prime factorizations.

Original entry on oeis.org

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

A066636 a(n) = A066638(n)/n, where A066638(n) is the smallest power of a squarefree kernel of n that is a multiple of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 11, 5, 1, 1, 27, 1, 2, 1, 13, 1, 4, 1, 49, 1, 1, 1, 15, 1, 1, 7, 1, 1, 1, 1, 17, 1, 1, 1, 3, 1, 1, 3, 19, 1, 1, 1, 125, 1, 1, 1, 21, 1

Offset: 1

Reinhard Zumkeller, Jan 09 2002

a(n) is the least m such that the prime power exponents of m*n are all equal; see also A062760. - David James Sycamore, Jun 13 2024

			12 = 2^2*3^1 so m = 3 (3*12 = 36 = 2^2*3^2).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A007947, A051903, A066638.

Cf. A062760.

Array[Apply[Times, #2[[All, 1]]]^Max[#2[[All, -1]] ]/#1 & @@ {#, FactorInteger@ #} &, 85] (* Michael De Vlieger, Nov 20 2017 *)

A066638(n) = { if(n==1, return(1)); my(f=factor(n),me=vecmax(f[, 2])); (prod(i=1, #f~, f[i, 1])^me); }; \\ After Charles R Greathouse IV's code.
A066636(n) = (A066638(n)/n); \\ Antti Karttunen, Nov 20 2017

a(n) = (A007947(n)^A051903(n))/n. - Antti Karttunen, Nov 20 2017

A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(n)).

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, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 10, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A034448, A322318, A323159.

Differs from A062760 for the first time at n=36, where a(36) = 2, while A062760(36) = 1.

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323163(n) = gcd(A003557(n), A034448(n));

a(n) = gcd(A003557(n), A034448(n)).

A295417 Self-inverse permutation of natural numbers: in prime factorization of n replace each positive prime exponent e with max + min - e, where max = A051903(n) and min = A051904(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 13, 14, 15, 16, 17, 12, 19, 50, 21, 22, 23, 54, 25, 26, 27, 98, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 49, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 64, 65, 66

Offset: 1

Rémy Sigrist, Nov 22 2017

nonn

This sequence was inspired by A293448.
This sequence first differs from A293448 at n = 42: a(42) = 42 whereas A293448(42) = 70.
a(A293448(n)) = A293448(a(n)) for any n > 0.
a(n) = n iff n belongs to A072774.
f(n) = f(a(n)) for any n > 0 and f in { A001221, A006530, A007947, A020639, A051903, A051904 }.
The lines visible in the logarithmic scatterplot of the sequence seems to correspond to integer sets where the function A062760 is constant (see logarithmic scatterplot in Links section).

			For n = 1620:
- 1620 = 2^2 * 3^4 * 5,
- A051903(1620) = 4 and A051904(1620) = 1,
- a(1620) = 2^(4+1-2) * 3^(4+1-4) * 5^(4+1-1) = 2^3 * 3 * 5^4 = 15000.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 1000000 terms (where the color is function of A062760(n))
Index entries for sequences that are permutations of the natural numbers

Cf. A001221, A006530, A007947, A020639, A051903, A051904, A062760, A072774 (fixed points), A293448.

a(n) = { my(f=factor(n)); if(#f~<=1, return(n), my(mi=vecmin(f[,2]), ma=vecmax(f[,2])); return(prod(i=1, #f~, f[i,1]^(ma+mi-f[i,2])))) }

a(n) = A007947(n)^(A051903(n) + A051904(n)) / n.

cp's OEIS Frontend

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

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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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A066636 a(n) = A066638(n)/n, where A066638(n) is the smallest power of a squarefree kernel of n that is a multiple of n.

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A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(n)).

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A295417 Self-inverse permutation of natural numbers: in prime factorization of n replace each positive prime exponent e with max + min - e, where max = A051903(n) and min = A051904(n).

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