A175068 a(n) = product of perfect divisors of n.

1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 12, 13, 14, 15, 128, 17, 18, 19, 20, 21, 22, 23, 24, 125, 26, 81, 28, 29, 30, 31, 64, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 343, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4096, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Jaroslav Krizek, Jan 23 2010

Keywords

• nonn

Comments

A perfect divisor d of n is a divisor such that d^k = n for some k >= 1.

Examples

			For n = 8: a(8) = 16; there are two perfect divisors of 8: 2 and 8; their product is 16.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 1000 terms from Harvey P. Dale)
• Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum 2 (1951), 254-260. See omega(n).

Crossrefs

Cf. A001597, A175069, A175084, A175085, A175087.

Cf. A000203, A346403, A253641.

Programs

Maple

A175068 := proc(n) local a,d,k ; if n = 1 then return 1; end if; a := 1 ; for d in numtheory[divisors](n) minus {1} do for k from 1 do if d^k = n then a := a*d ; end if; if d^k >= n then break; end if; end do: end do: a ; end proc:
seq(A175068(n),n=1..80) ; # R. J. Mathar, Apr 14 2011

Mathematica

Table[Times@@Select[Rest[Divisors[n]],IntegerQ[Log[#,n]]&],{n,70}] (* Harvey P. Dale, May 01 2017 *)

PARI

A175068(n) = { my(m=1); fordiv(n,d,if((1==d)||(d^valuation(n,d))==n,m*=d)); (m); }; \\ Antti Karttunen, Nov 21 2017

Formula

a(n) > n for perfect powers n = A001597(m) for m > 2.

a(n) = A175069(n) * n. - Jaroslav Krizek, Jan 24 2010

From Ridouane Oudra, Nov 23 2024: (Start)

a(n) = n, for n in A007916.

a(n^m) = n^sigma(m), for n in A007916 and m an integer.

More generally, for all integer n we have :

a(n) = n^(sigma(A253641(n))/A253641(n)).

a(n) = n^(A346403(n)/A253641(n)). (End)