A057564 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]].

666, 19674, 309114, 369594, 715194, 1180026, 1924794, 2722266, 2741274, 4261914, 5564826, 6296634, 6623226, 6826266, 7206906, 9726648, 15956154, 16046874, 16478874, 19263546, 25333146, 27706554, 28151514, 32938074

Offset: 1

Author

Douglas E. Iannucci, Oct 04 2000

Keywords

• easy
• nonn

Comments

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).

Examples

			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.
		

References

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

Programs

Mathematica

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 *)