A079870 -id:A079870 - 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

A079871 a(1)=1 and for n>1: ceiling(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, 9, 7, 8, 9, 16, 11, 27, 13, 16, 16, 16, 17, 27, 19, 27, 25, 25, 23, 81, 25, 36, 27, 64, 29, 64, 31, 32, 36, 36, 36, 81, 37, 49, 49, 81, 41, 64, 43, 64, 64, 49, 47, 243, 49, 64, 64, 64, 53, 81, 64, 81, 64, 64, 59, 81, 61, 64, 64, 64, 81, 125, 67, 125, 81, 125, 71

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001222, A006530, A020639, A068794, A068795, A079869, A079870.

A079871[n_] := If [n == 1, 1, Ceiling[n^(1/#)]^# & [PrimeOmega[n]]];
Array[A079871, 100] (* Paolo Xausa, Oct 27 2024 *)

a(n) = if (n==1, 1, ceil(n^(1/bigomega(n)))^bigomega(n)); \\ Michel Marcus, May 31 2016

a(n) = A079870(n)^A001222(n).
a(n) >= A079869(n); A020639(n) <= a(n) <= A006530(n);
a(m) = m = A079869(m) iff m is a prime power (A000961).

A273283 Least prime not less than the geometric mean of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 5, 5, 2, 17, 3, 19, 3, 5, 5, 23, 3, 5, 7, 3, 5, 29, 5, 31, 2, 7, 7, 7, 3, 37, 7, 7, 3, 41, 5, 43, 5, 5, 7, 47, 3, 7, 5, 11, 5, 53, 3, 11, 3, 11, 11, 59, 3, 61, 11, 5, 2, 11, 5, 67, 5, 11, 5, 71, 3, 73, 11, 5, 5, 11, 5, 79, 3, 3, 11

Offset: 2

Giuseppe Coppoletta, May 19 2016

nonn

A079870(n) <= a(n) <= A006530(n) <= n and a(n) = n iff n is prime, while a(n)= A079870(n) iff A079870(n) is prime.

			a(46)=7 because 7 is the least prime not less than sqrt(2*23).
a(84)=5 and A273282(84)=3 because A001222(84)=4 and 3 < 84^(1/4) < 5.

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

Cf. A273282, A273285, A273289, A079870, A079871, A006530, A001222.

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

[next_prime(ceil(n^(1/sloane.A001222(n)))-1) for n in (2..82)]

For n >= 2, a(n) = A007918(A079870(n)).

A273289 Least prime not less than the median of all prime divisors of n counted with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 2, 13, 5, 5, 2, 17, 3, 19, 2, 5, 7, 23, 2, 5, 11, 3, 2, 29, 3, 31, 2, 7, 11, 7, 3, 37, 11, 11, 2, 41, 3, 43, 2, 3, 13, 47, 2, 7, 5, 11, 2, 53, 3, 11, 2, 11, 17, 59, 3, 61, 17, 3, 2, 11, 3, 67, 2, 13, 5, 71, 2, 73, 23, 5, 2, 11, 3, 79, 2, 3, 23

Offset: 2

Giuseppe Coppoletta, May 25 2016

nonn

A273288(n)<= a(n)<= A006530<= n and a(n) = n iff n is prime.

			a(76) = 2 because the median of its prime factors [2, 2, 19] is the central value 2 (and it is prime).
a(308) = 5 because the median of [2, 2, 7, 11] is commonly defined as the mean of the central values (2+7)/2 = 4.5 and the next prime is 5.

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

Cf. A273283, A273285, A273288, A273291, A079870, A006530.

Table[If[PrimeQ@ #, #, NextPrime@ #] &@ Median@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ n, 1], {n, 2, 82}] (* Michael De Vlieger, May 27 2016 *)

r = lambda n: [f[0] for f in factor(n) for _ in range(f[1])]; [next_prime(ceil(median(r(n)))-1) for n in (2..100)]

A079868 a(1)=1 and for n>1: round(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 3, 19, 3, 5, 5, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 6, 6, 6, 2, 37, 6, 6, 3, 41, 3, 43, 4, 4, 7, 47, 2, 7, 4, 7, 4, 53, 3, 7, 3, 8, 8, 59, 3, 61, 8, 4, 2, 8, 4, 67, 4, 8, 4, 71, 2, 73, 9, 4, 4, 9, 4, 79, 2, 3, 9, 83, 3, 9, 9, 9, 3, 89, 3, 10, 5, 10, 10

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

A079866(n)<=a(n)<=A079870(n); A020639(n)<=a(n)<=A006530(n);

a(m)=A079866(m)=A079870(m) iff m is a prime power (A000961).

Cf. A079869(n)=a(n)^A001222(n), A079881.

Join[{1},Table[Floor[n^(1/PrimeOmega[n])+1/2],{n,2,100}]] (* Harvey P. Dale, Aug 11 2012 *)

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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A079871 a(1)=1 and for n>1: ceiling(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

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A273283 Least prime not less than the geometric mean of all prime divisors of n counted with multiplicity.

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A273289 Least prime not less than the median of all prime divisors of n counted with multiplicity.

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A079868 a(1)=1 and for n>1: round(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

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