%I #8 Aug 04 2020 11:48:37

%S 666,19674,309114,369594,715194,1180026,1924794,2722266,2741274,

%T 4261914,5564826,6296634,6623226,6826266,7206906,9726648,15956154,

%U 16046874,16478874,19263546,25333146,27706554,28151514,32938074

%N Natural numbers n for which the following three properties hold: (i) Phi[Sigma[n]]=2*phi(n), (ii) Sigma[n-2]=2*Sigma[Phi[n-2]], (iii) Sigma[n+2]=2*Sigma[Phi[n+2]].

%C Called "Beastly neighbors" because 666 is the least element of the sequence and because these numbers are seen to have a symmetric relationship with their two "neighbors," using analogy to numbering houses on the same side of the street (e.g. 664 and 668 are "neighbors" of 666).

%D D. Iannucci, "The neighbors of the Beast," Journal of Recreational Mathematics, Volume 31, Number 1, 2002, pp. 52-55.

%e Phi[Sigma[666]]=Phi[1482]=432=2*216=2*Phi[666], Sigma[664]=1260=2*630=2*Sigma[328]=Sigma[Phi[664]], Sigma[668]=1176=2*588=2*Sigma[332]=Sigma[Phi[668]]. Thus 666 is an element of the sequence.

%t Select[Range[33*10^6],EulerPhi[DivisorSigma[1,#]]==2EulerPhi[#]&&DivisorSigma[ 1,#-2]==2DivisorSigma[1,EulerPhi[#-2]]&&DivisorSigma[ 1,#+2] == 2DivisorSigma[1,EulerPhi[#+2]]&] (* _Harvey P. Dale_, Aug 04 2020 *)

%K easy,nonn

%O 1,1

%A Douglas E. Iannucci, Oct 04 2000