A065655 Composite numbers k such that sigma(k)*(phi(k) + 2) is a square.

28, 90, 156, 184, 374, 1855, 2162, 2170, 2280, 2376, 2415, 2665, 3160, 4970, 5270, 5740, 6402, 6494, 7414, 8400, 9118, 10656, 11155, 12400, 14632, 14910, 15010, 15906, 18183, 18792, 22648, 24645, 24734, 24920, 25844, 26670, 27478, 28990

Offset: 1

Author

Labos Elemer, Nov 12 2001

Keywords

• nonn

Examples

			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.
For k = 28, sigma(28) = 56, phi(28) = 12, 56*(12 + 2) = 784 = 28*28, so 28 is a composite solution.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2500 (terms 1..500 from Harry J. Smith)

Crossrefs

Cf. A000010, A000203.

Programs

Mathematica

Select[Range@ 30000, Function[n, And[CompositeQ@ n, IntegerQ@ Sqrt[# EulerPhi@ n + 2 #] &@ DivisorSigma[1, n]]]] (* Michael De Vlieger, Mar 18 2017 *)

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