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%I #28 Apr 25 2020 22:11:11
%S 1,2,3,8,5,6,7,16,27,10,11,12,13,14,15,128,17,18,19,20,21,22,23,24,
%T 125,26,81,28,29,30,31,64,33,34,35,216,37,38,39,40,41,42,43,44,45,46,
%U 47,48,343,50,51,52,53,54,55,56,57,58,59,60,61,62,63,4096,65,66,67,68,69,70
%N a(n) = product of perfect divisors of n.
%C A perfect divisor d of n is a divisor such that d^k = n for some k >= 1.
%H Antti Karttunen, <a href="/A175068/b175068.txt">Table of n, a(n) for n = 1..16384</a> (first 1000 terms from Harvey P. Dale)
%H Jan Mycielski, <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2140.pdf">Sur les représentations des nombres naturels par des puissances à base et exposant naturels</a>, Colloquium Mathematicum 2 (1951), 254-260. See omega(n).
%F a(n) > n for perfect powers n = A001597(m) for m > 2.
%F a(n) = A175069(n) * n. - _Jaroslav Krizek_, Jan 24 2010
%e For n = 8: a(8) = 16; there are two perfect divisors of 8: 2 and 8; their product is 16.
%p 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:
%p seq(A175068(n),n=1..80) ; # _R. J. Mathar_, Apr 14 2011
%t Table[Times@@Select[Rest[Divisors[n]],IntegerQ[Log[#,n]]&],{n,70}] (* _Harvey P. Dale_, May 01 2017 *)
%o (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
%Y Cf. A001597, A175069, A175084, A175085, A175087.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Jan 23 2010
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