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%I #20 Sep 08 2022 08:45:42
%S 1,2,3,8,5,6,7,16,27,10,11,72,13,14,15,128,17,108,19,200,21,22,23,144,
%T 125,26,81,392,29,30,31,64,33,34,35,46656,37,38,39,400,41,42,43,968,
%U 675,46,47,3456,343,500,51,1352,53,324,55,784,57,58,59,1800,61,62,1323,4096
%N a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.
%C The exponential divisors of a number n = Product p(i)^e(i) are all numbers of the form Product p(i)^s(i) where s(i) divides e(i) for all i.
%C Not multiplicative: a(3)=3 (e-divisor 3^1), a(4)=8 (e-divisors 2^1 and 2^2), but a(12)=72 (e-divisors 3*2 and 3*2^2) <> a(3)*a(4). - _R. J. Mathar_, Apr 14 2011
%H Amiram Eldar, <a href="/A157488/b157488.txt">Table of n, a(n) for n = 1..10000</a>
%H József Sándor, <a href="http://fs.unm.edu/SN/NoteExponentialDivisors.pdf">A note on exponential divisors and related arithmetic functions</a>, Scientia Magna, Vol. 1, No. 1 (2005), pp. 97-101.
%F a(1) = 1, a(p) = p, a(p*q) = p*q, a(p*q...*z) = pq...z, a(p^k) = p^(A000203(k)), for p, q, ..., z distinct primes and k > 1 an integer.
%F From _Amiram Eldar_, Jun 03 2020: (Start)
%F If n = Product_{i} p_i^e_i then a(n) = Product_{i} p_i^(sigma(e_i) * d_exp(n) / d(e_i)), where d_exp(n) = Product_{i} d(e_i) is the number of exponential divisors of n (A049419), d(e) and sigma(e) are the number of divisors (A000005) of e and their sum (A000203).
%F a(n) <= A007955(n) with equality if and only if n is noncomposite. (End)
%e For n = 16 = 2^4 = the a(16) = 2^(A000203(4)) = 2^7 = 128. e-divisors of number 16 is 2, 4, 16, their product is 128.
%t f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] :=(Times @@ (f @@@ (fct = FactorInteger[n])))^(Times @@ DivisorSigma[0, Last /@ fct]); Array[a, 100] (* _Amiram Eldar_, Jun 03 2020 *)
%o (Magma) cat [ &*[ d: d in Divisors(n) | forall(t) { p: p in P | v gt 0 and e mod v eq 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..64] ]; // _Klaus Brockhaus_, May 26 2009
%Y Cf. A000005, A000027, A000040, A000203, A006881, A000961, A049419, A051377, A120944, A322791.
%Y Similar sequences: A007955, A061537, A274029.
%K nonn
%O 1,2
%A _Jaroslav Krizek_, Mar 01 2009
%E a(1) = 1 from _N. J. A. Sloane_, Mar 02 2009
%E a(60) corrected by _Klaus Brockhaus_, May 26 2009
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