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

1,2

Comments

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

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).

Formula

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A138682 A065632 A242480 * A355264 A157488 A188385

Adjacent sequences: A175065 A175066 A175067 * A175069 A175070 A175071

Keyword

nonn

Author

Jaroslav Krizek, Jan 23 2010

Status

approved