A055008 Numbers k such that gcd(phi(k), sigma(k)) = 1 with phi = A000010, sigma = A000203.

1, 2, 4, 8, 9, 16, 25, 32, 36, 50, 64, 81, 100, 121, 128, 144, 225, 242, 256, 289, 324, 400, 484, 512, 529, 576, 578, 625, 729, 800, 841, 900, 1024, 1058, 1089, 1156, 1250, 1296, 1600, 1681, 1682, 1936, 2025, 2048, 2116, 2209, 2304, 2312, 2401, 2500, 2601

Offset: 1

Author

Labos Elemer, May 31 2000

Keywords

• nonn

Comments

The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020

Conjecture: Every term is a square or twice a square. - Jason Yuen, May 16 2024

The conjecture is true: If k is neither a square nor twice a square (i.e., in A028983), then sigma(k) is even. Since gcd(phi(k), sigma(k)) = 1, then phi(k) must be odd, but phi(k) is odd only for k = 1 and 2. - Amiram Eldar, May 19 2024

Examples

			For n = 484, phi(484) = 220 = 2*2*5*11, sigma(484) = 931 = 7*7*19, and gcd(220,931) = 1.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.

Crossrefs

Cf. A000010, A000203, A009223, A028983.

Programs

Mathematica

Select[Range@ 2700, CoprimeQ[EulerPhi@ #, DivisorSigma[1, #]] &] (* Michael De Vlieger, Feb 05 2017 *)
Select[With[{max = 51}, Union[Array[#^2 &, max], Array[2*#^2 &, Floor[max / Sqrt[2]]]]], CoprimeQ[EulerPhi[#], DivisorSigma[1, #]] &] (* Amiram Eldar, May 19 2024 *)

PARI

is(n)=gcd(sigma(n),eulerphi(n))==1 \\ Charles R Greathouse IV, Feb 19 2013

Extensions

Incorrect comment removed by Charles R Greathouse IV, Feb 19 2013