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A067911 Product of gcd(k,n) for 1 <= k <= n.

%I A067911 #40 Nov 27 2021 20:01:42
%S A067911 1,2,3,8,5,72,7,128,81,800,11,41472,13,6272,30375,32768,17,3359232,19,
%T A067911 20480000,750141,247808,23,13759414272,15625,1384448,1594323,
%U A067911 5035261952,29,30233088000000,31,2147483648,235782657,37879808
%N A067911 Product of gcd(k,n) for 1 <= k <= n.
%H A067911 T. D. Noe, <a href="/A067911/b067911.txt">Table of n, a(n) for n = 1..500</a>
%H A067911 Johann Cigler, <a href="https://arxiv.org/abs/2006.06242">Some remarks on the power product expansion of the q-exponential series</a>, arXiv:2006.06242 [math.CO], 2020.
%H A067911 Gottfried Helms, <a href="http://go.helms-net.de/math/musings/dreamofasequence.pdf">A dream of a (number-) sequence</a>, 2007-2009.
%H A067911 L. Toth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html">A survey of gcd-sum functions</a>, J. Int. Seq. 13 (2010) # 10.8.1.
%F A067911 a(n) = Product_{d|n} d^phi(n/d). - _Vladeta Jovovic_, Mar 08 2004
%F A067911 a(n) = n*A051190(n). - _Peter Luschny_, Apr 07 2013
%F A067911 a(n) = Product_{k=1..n} (n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))) where phi = A000010. - _Richard L. Ollerton_, Nov 07 2021
%p A067911 with(numtheory): a := n -> mul(d^phi(n/d), d = divisors(n)):
%p A067911 seq(a(i), i = 1..34); # _Peter Luschny_, Apr 07 2013
%t A067911 a[n_] := Product[d^EulerPhi[n/d], {d, Divisors[n]}];
%t A067911 Array[a, 34] (* _Jean-François Alcover_, Jun 03 2019 *)
%o A067911 (Sage)
%o A067911 A067911 = lambda n: mul(gcd(n,i) for i in range(n))
%o A067911 [A067911(n) for n in (1..34)] # _Peter Luschny_, Apr 07 2013
%o A067911 (PARI) a(n) = prod(k=1, n, gcd(k, n)); \\ _Michel Marcus_, Aug 23 2016
%Y A067911 Cf. A051190, A051696.
%Y A067911 In A018804 the product is replaced by sum.
%Y A067911 Product of terms in n-th row of A050873.
%Y A067911 Cf. A000010 (comments on product formulas).
%K A067911 nonn
%O A067911 1,2
%A A067911 Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002
%E A067911 Extended and edited by _John W. Layman_, Mar 14 2002