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%I #21 Mar 27 2022 03:23:48
%S 1,6,30,66,102,210,270,318,330,420,462,510,546,570,642,672,690,714,
%T 840,870,924,930,966,1122,1320,1410,1428,1518,1590,1638,1722,1770,
%U 1890,1932,2130,2226,2280,2310,2346,2370,2670,2730,2760,2838,2970,2982,3102,3162,3210,3360,3444,3486,3498,3570,3720,3780,3948,3990
%N Numbers k such that number of divisors of k and sum of divisors of k divides product of divisors of k and the average of the divisors of k is an integer.
%C Intersection of A003601, A120736 and A145551.
%C Numbers k such that A000005(k)|A007955(k), A000203(k)|A007955(k) and A000005(k)| A000203(k).
%C Numbers k such that A000005(k)|A062981(k), A072861(k)|A062758(k) and A245656(k) = 1.
%H Amiram Eldar, <a href="/A277521/b277521.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A277521/a277521.jpg">Plot of a(n)/n^(3/2) for n = 1..20000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_number">Arithmetic number</a>.
%e a(2) = 6 because 6 has 4 divisors {1,2,3,6}, 1*2*3*6/4 = 9, 1*2*3*6/(1 + 2 + 3 + 6) = 3 and (1 + 2 + 3 + 6)/4 = 3 are integer.
%p with(numtheory): P:=proc(q) local a,b,k,n;for n from 1 to q do
%p a:=divisors(n); b:=mul(a[k],k=1..nops(a));
%p if type(sigma(n)/tau(n),integer) and type(b/sigma(n),integer) and
%p type(b/tau(n),integer) then print(n); fi;
%p od; end: P(10^5); # _Paolo P. Lava_, Oct 20 2016
%t Select[Range[4000], Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[1, #1]] && Divisible[Sqrt[#1]^DivisorSigma[0, #1], DivisorSigma[0, #1]] && Divisible[DivisorSigma[1, #1], DivisorSigma[0, #1]] & ]
%Y Cf. A000005, A000203, A003601, A007955, A062758, A062981, A072861, A120736, A145551, A245656.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Oct 19 2016
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