A079867 -id:A079867 - cp's OEIS frontend

A079866 a(1)=1 and for n>1: floor(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 4, 4, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 6, 6, 2, 41, 3, 43, 3, 3, 6, 47, 2, 7, 3, 7, 3, 53, 2, 7, 2, 7, 7, 59, 2, 61, 7, 3, 2, 8, 4, 67, 4, 8, 4, 71, 2, 73, 8, 4, 4, 8, 4, 79, 2, 3, 9, 83, 3, 9, 9, 9, 3, 89, 3, 9, 4, 9, 9, 9, 2

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

a(n) <= A079868(n).
A020639(n) <= a(n) <= A006530(n);
a(m) = A079868(m) = A079870(m) iff m is a prime power (A000961).

David W. Wilson, Table of n, a(n) for n = 1..10000

A079867(n) = a(n)^A001222(n).

Cf. A079747, A358890.

A079866 := proc(n)
    root[numtheory[bigomega](n)](n) ;
    floor(%) ;
end proc:
seq(A079866(n),n=1..97) ; # R. J. Mathar, Sep 07 2016

Join[{1}, Table[Floor[n^(1/PrimeOmega[n])], {n, 2, 20}]] (* G. C. Greubel, Sep 16 2016 *)

a(n) = if (n==1, 1, sqrtnint(n, bigomega(n))); \\ Michel Marcus, Sep 09 2016

A079869 a(1)=1 and for n>1: round(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 16, 16, 16, 17, 27, 19, 27, 25, 25, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 36, 36, 36, 16, 37, 36, 36, 81, 41, 27, 43, 64, 64, 49, 47, 32, 49, 64, 49, 64, 53, 81, 49, 81, 64, 64, 59, 81, 61, 64, 64, 64, 64, 64, 67, 64, 64, 64, 71, 32, 73

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

A079867(n)<=a(n)<=A079869(n); A020639(n)<=a(n)<=A006530(n);

a(m)=m=A079867(m)=A079871(m) iff m is a prime power (A000961).

a(n)=A079868(n)^A001222(n). Cf. A079872, A068794, A068795.

ron[n_]:=Module[{c=PrimeOmega[n]},Round[n^(1/c)]^c]; Join[{1},Array[ ron,80,2]] (* Harvey P. Dale, Jun 17 2020 *)

A273284 A273282(n)^Omega(n), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 9, 9, 16, 17, 8, 19, 8, 9, 9, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 25, 25, 25, 16, 37, 25, 25, 16, 41, 27, 43, 27, 27, 25, 47, 32, 49, 27, 49, 27, 53, 16, 49, 16, 49, 49, 59, 16, 61, 49, 27, 64, 49, 27, 67, 27, 49, 27, 71, 32, 73, 49

Offset: 2

Giuseppe Coppoletta, May 20 2016

nonn

a(n) <= A079867(n) for any n>=2.

a(n) = n iff n is the power of a prime (A246655).

			a(33) = 25 because Omega(33)=2 and 5^2 < 33 < 7^2.
If n= 3^3 * 31^2 * 67 then a(n)= 7^6 and A273285(n)=11^6 because Omega(n)=6 and 7^6 < n < 11^6.

Giuseppe Coppoletta, Table of n, a(n) for n = 2..10000

Cf. A273282, A273285, A273290, A079867, A246655, A001222.

Table[NextPrime[(Floor[n^(1/PrimeOmega[n])] + 1) , -1]^PrimeOmega[n], {n,2,50}] (* G. C. Greubel, May 26 2016 *)

a(n) = my(bn=bigomega(n)); precprime(sqrtnint(n, bn))^bn; \\ Michel Marcus, May 24 2016

s=sloane.A001222; [previous_prime(floor(n^(1/s(n)))+1)^s(n) for n in (2..74)]

a(n) = A273282(n)^A001222(n).

A273290 A273288(n)^Omega(n), where Omega = A001222.

Original entry on oeis.org

2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 9, 9, 16, 17, 27, 19, 8, 25, 25, 23, 16, 25, 49, 27, 8, 29, 27, 31, 32, 49, 49, 25, 16, 37, 49, 49, 16, 41, 27, 43, 8, 27, 121, 47, 32, 49, 125, 49, 8, 53, 81, 49, 16, 121, 169, 59, 16, 61, 169, 27, 64, 49, 27, 67, 8, 169, 125

Offset: 2

Giuseppe Coppoletta, May 25 2016

nonn

a(n) is by definition the power of a prime. It coincides with n iff n is the power of a prime (A246655).

a(n) <= A273291(n)

			a(70) = A273291(70) = 5^3 because the median  of its prime factors [2, 5, 7] is the central value 5 (prime) and Omega(70)=3.
a(308) = 3^4 because Omega(308)=4 and the median of [2, 2, 7, 11] is (2+7)/2 = 4.5, whose previous prime is 3.

Giuseppe Coppoletta, Table of n, a(n) for n = 2..10000

Cf. A273284, A273288, A273291, A079867, A246655, A001222.

Table[Prime[PrimePi@ Median@ #]^Length@ # &@ Flatten@ Apply[Table[#1 {#2}] &, FactorInteger@ n, 1], {n, 2, 75}] (* Michael De Vlieger, May 27 2016 *)

def pfwr(n): return flatten([([p] * m) for (p, m) in factor(n)]) # (list of prime factors of n with repetition)
[previous_prime(floor(median(pfwr(n)))+1)^sloane.A001222(n) for n in (2..70)]

a(n) = A273288(n)^A001222(n).

cp's OEIS Frontend

A079866 a(1)=1 and for n>1: floor(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

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A079869 a(1)=1 and for n>1: round(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

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A273284 A273282(n)^Omega(n), where Omega = A001222.

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A273290 A273288(n)^Omega(n), where Omega = A001222.

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