A067911 - OEIS

%I #40 Nov 27 2021 20:01:42

%S 1,2,3,8,5,72,7,128,81,800,11,41472,13,6272,30375,32768,17,3359232,19,

%T 20480000,750141,247808,23,13759414272,15625,1384448,1594323,

%U 5035261952,29,30233088000000,31,2147483648,235782657,37879808

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

%H T. D. Noe, <a href="/A067911/b067911.txt">Table of n, a(n) for n = 1..500</a>

%H 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 Gottfried Helms, <a href="http://go.helms-net.de/math/musings/dreamofasequence.pdf">A dream of a (number-) sequence</a>, 2007-2009.

%H 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 a(n) = Product_{d|n} d^phi(n/d). - _Vladeta Jovovic_, Mar 08 2004

%F a(n) = n*A051190(n). - _Peter Luschny_, Apr 07 2013

%F 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 with(numtheory): a := n -> mul(d^phi(n/d), d = divisors(n)):

%p seq(a(i), i = 1..34); # _Peter Luschny_, Apr 07 2013

%t a[n_] := Product[d^EulerPhi[n/d], {d, Divisors[n]}];

%t Array[a, 34] (* _Jean-François Alcover_, Jun 03 2019 *)

%o (Sage)

%o A067911 = lambda n: mul(gcd(n,i) for i in range(n))

%o [A067911(n) for n in (1..34)] # _Peter Luschny_, Apr 07 2013

%o (PARI) a(n) = prod(k=1, n, gcd(k, n)); \\ _Michel Marcus_, Aug 23 2016

%Y Cf. A051190, A051696.

%Y In A018804 the product is replaced by sum.

%Y Product of terms in n-th row of A050873.

%Y Cf. A000010 (comments on product formulas).

%K nonn

%O 1,2

%A Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

%E Extended and edited by _John W. Layman_, Mar 14 2002