A065655 Composite numbers n such that sigma(n)*phi(n) + 2*sigma(n) 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

1,1

Links

Harry J. Smith, Table of n, a(n) for n = 1..500

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A000010, A000203.

Sequence in context: A042546 A164689 A096384 * A341623 A272399 A113958

Adjacent sequences: A065652 A065653 A065654 * A065656 A065657 A065658

Keyword

nonn

Author

Labos Elemer, Nov 12 2001

Status

approved