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A160199 Product of non-exponential divisors of n.

%I A160199 #18 Feb 16 2025 08:33:10
%S A160199 1,1,1,1,1,6,1,4,1,10,1,24,1,14,15,8,1,54,1,40,21,22,1,2304,1,26,9,56,
%T A160199 1,27000,1,512,33,34,35,216,1,38,39,6400,1,74088,1,88,135,46,1,73728,
%U A160199 1,250,51,104,1,26244,55,12544,57,58,1,25920000,1,62,189,512,65,287496,1
%N A160199 Product of non-exponential divisors of n.
%C A160199 The non-exponential divisors of n are those divisors of n that are not exponential divisors of n.
%C A160199 There are only a few duplicates > 1. For example a(32) = a(64) = 512, a(243) = a(729) = 19683, a(3125) = a(15625) = 1953125. _Antti Karttunen_, Jan 24 2025
%H A160199 Antti Karttunen, <a href="/A160199/b160199.txt">Table of n, a(n) for n = 1..25000</a>
%H A160199 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>
%F A160199 a(n) = A007955(n) / A157488(n).
%e A160199 The divisors of 6 are 1, 2, 3, 6. The only exponential divisor of 6 is 6, hence a(6) = 1*2*3 = 6.
%e A160199 The divisors of 16 are 1, 2, 4, 8, 16. The exponential divisors of 16 are 2, 4, 16, hence a(16) = 1*8 = 8.
%t A160199 f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] := Module[{fct = FactorInteger[n], e}, e = fct[[;; , 2]]; n^(Times @@ (e + 1)/2)/(Times @@ (f @@@ fct))^(Times @@ DivisorSigma[0, e])]; Array[a, 100] (* _Amiram Eldar_, Jan 25 2025 *)
%o A160199 (Magma) [1] cat [ &*[ d: d in Divisors(n) | exists(t) { p: p in P | v eq 0 or e mod v gt 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..67] ]; // _Klaus Brockhaus_, May 26 2009
%o A160199 (PARI)
%o A160199 A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2));
%o A160199 A049419(n) = factorback(apply(numdiv,factor(n)[,2]));
%o A160199 A157488(n) = { my(f=factor(n), dexp=A049419(n)); prod(i=1, #f~, f[i, 1]^((sigma(f[i, 2]) * dexp / numdiv(f[i, 2])))); };
%o A160199 A160199(n) = (A007955(n) / A157488(n)); \\ _Antti Karttunen_, Jan 24 2025
%Y A160199 Cf. A007955 (product of divisors of n), A157488 (product of exponential divisors of n), A049419.
%K A160199 nonn
%O A160199 1,6
%A A160199 _Jaroslav Krizek_, May 04 2009
%E A160199 Edited by _Klaus Brockhaus_, May 26 2009