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Composite numbers k such that sigma(k)*(phi(k) + 2) is a square.
5

%I #30 Nov 13 2024 08:30:28

%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 k such that sigma(k)*(phi(k) + 2) is a square.

%H Amiram Eldar, <a href="/A065655/b065655.txt">Table of n, a(n) for n = 1..2500</a> (terms 1..500 from Harry J. Smith)

%e Since for a prime p, sigma(p)*(phi(p) + 2) = (p+1)*((p-1) + 2) = (p+1)^2 is a square, all primes are solutions.

%e For k = 28, sigma(28) = 56, phi(28) = 12, 56*(12 + 2) = 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

%o (PARI) is(k) = if(k == 1 || isprime(k), 0, my(f = factor(k), s = sigma(f), p = eulerphi(f)); issquare(s * (p+2))); \\ _Amiram Eldar_, Nov 12 2024

%Y Cf. A000010, A000203.

%K nonn,changed

%O 1,1

%A _Labos Elemer_, Nov 12 2001