A068794 -id:A068794

A068795

In prime factorization of n replace all primes with the greatest prime factor of n; a(1)=1.

+10
9

1, 2, 3, 4, 5, 9, 7, 8, 9, 25, 11, 27, 13, 49, 25, 16, 17, 27, 19, 125, 49, 121, 23, 81, 25, 169, 27, 343, 29, 125, 31, 32, 121, 289, 49, 81, 37, 361, 169, 625, 41, 343, 43, 1331, 125, 529, 47, 243, 49, 125, 289, 2197, 53, 81, 121, 2401, 361, 841, 59, 625, 61

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..61.

FORMULA

a(n) = A006530(n)^A001222(n).

A068794(n) <= n <= a(n).

a(n) = A068794(n) iff n = 1 or n = p^k for some prime p, k > 0.

EXAMPLE

a(30) = a(2*3*5) = 5*5*5 = 125.

MAPLE

with(NumberTheory): A068795 := n -> max(PrimeFactors(n))^Omega(n):

seq(A068795(n), n = 1..61); # Peter Luschny, Oct 25 2024

CROSSREFS

Cf. A001222, A006530, A068794.

KEYWORD

nonn,changed

AUTHOR

Reinhard Zumkeller, Mar 05 2002

STATUS

approved

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).

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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).

MATHEMATICA

A079871[n_] := If [n == 1, 1, Ceiling[n^(1/#)]^# & [PrimeOmega[n]]];

Array[A079871, 100] (* Paolo Xausa, Oct 27 2024 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,changed

AUTHOR

Reinhard Zumkeller, Jan 13 2003

STATUS

approved

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).

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..73.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jan 13 2003

STATUS

approved

A079870

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

A079871(n) = a(n)^A001222(n).

a(n) >= A079868(n); A020639(n) <= a(n) <= A006530(n);

a(m) = A079868(m) iff m is a prime power (A000961).

MATHEMATICA

A079870[n_] := If [n == 1, 1, Ceiling[n^(1/PrimeOmega[n])]];

Array[A079870, 100] (* Paolo Xausa, Oct 28 2024 *)

PROG

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

CROSSREFS

Cf. A001222, A006530, A020639, A068794, A068795, A079869, A079871.

KEYWORD

nonn,changed

AUTHOR

Reinhard Zumkeller, Jan 13 2003

STATUS

approved

A079867

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

a(m)=m=A079869(m)=A079871(m) iff m is a prime power (A000961).

LINKS

Table of n, a(n) for n=1..74.

MATHEMATICA

Join[{1}, Table[Floor[n^(1/PrimeOmega[n])]^PrimeOmega[n], {n, 2, 80}]] (* Harvey P. Dale, May 19 2018 *)

CROSSREFS

a(n)=A079866(n)^A001222(n), cf. A068794, A068795.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jan 13 2003

STATUS

approved

A079881

mpf(n)^Omega(n), where mpf(n) is the median prime factor of n (A079879).

+10
2

1, 2, 3, 4, 5, 4, 7, 8, 9, 4, 11, 8, 13, 4, 9, 16, 17, 27, 19, 8, 9, 4, 23, 16, 25, 4, 27, 8, 29, 27, 31, 32, 9, 4, 25, 16, 37, 4, 9, 16, 41, 27, 43, 8, 27, 4, 47, 32, 49, 125, 9, 8, 53, 81, 25, 16, 9, 4, 59, 16, 61, 4, 27, 64, 25, 27, 67, 8, 9, 125, 71, 32, 73, 4, 125, 8, 49, 27, 79, 32

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A068794(n)<=a(n)<=A068795(n);

a(m)=m=A068794(m)=A068795(m) iff m is a prime power (A000961).

LINKS

Table of n, a(n) for n=1..80.

CROSSREFS

a(n)=A079879(n)^A001222(n), A033676, O000004.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jan 13 2003

STATUS

approved