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%I #17 Mar 18 2017 11:28:04
%S 28,90,156,184,374,1855,2162,2170,2280,2376,2415,2665,3160,4970,5270,
%T 5740,6402,6494,7414,8400,9118,10656,11155,12400,14632,14910,15010,
%U 15906,18183,18792,22648,24645,24734,24920,25844,26670,27478,28990
%N Composite numbers n such that sigma(n)*phi(n) + 2*sigma(n) is a square.
%H Harry J. Smith, <a href="/A065655/b065655.txt">Table of n, a(n) for n = 1..500</a>
%e Since (p+1)(p-1) + 2(p+1) = p^2 + 2p + 1 = (p+1)^2 is a square, all primes are solutions. For n = 28, sigma(28) = 56, phi(28) = 12, 56*12 + 2*56 = 784 = 28*28, so 28 is a composite solution.
%t Select[Range@ 30000, Function[n, And[CompositeQ@ n, IntegerQ@ Sqrt[# EulerPhi@ n + 2 #] &@ DivisorSigma[1, n]]]] (* _Michael De Vlieger_, Mar 18 2017 *)
%o (PARI) { n=0; for (m=1, 10^9, if (isprime(m), next); s=sigma(m)*eulerphi(m) + 2*sigma(m); if (issquare(s), write("b065655.txt", n++, " ", m); if (n==500, return)) ) } \\ _Harry J. Smith_, Oct 25 2009
%Y Cf. A000010, A000203.
%K nonn
%O 1,1
%A _Labos Elemer_, Nov 12 2001
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