A055650 Numbers k such that k | phi(k)*d(k) - sigma(k), where phi=A000010, d=A000005 and sigma=A000203.

1, 3, 14, 42, 76, 376, 3608, 163712, 163944, 196128, 277688, 491136, 833064, 849120, 905814, 911008, 1080328, 1653520, 1847898, 1935128, 2733024, 3145216, 3240984, 4586240, 4734736, 4960560, 5805384, 13758720, 16582752, 25244956, 34961040, 38521440, 48177990, 56240352

Offset: 1

Author

Robert G. Wilson v, Jun 06 2000

Keywords

• nonn

Comments

From Farideh Firoozbakht, Mar 17 2007: (Start)

I. If p is an odd prime then m = 2^k*p is in the sequence iff p = (k+3)*2^k - 1. For example, 14, 76, 376, 163712, 3145216, 1073733632, 1443108749312 and 67185481812096157153425363042304 are such terms. The numbers k such that (k+3)*2^k - 1 is prime up to 10000 are 1, 2, 3, 7, 9, 13, 18, 50, 210, 301, 349, 1160, 1796, 2677 and 8823. Thus 2^8823*(8826*2^8823-1) is the largest such term that I have found.

II. If m is in the sequence and 3 | phi(m)*d(m) - sigma(m) but 3 doesn't divide m then 3*m is in the sequence. Thus 1, 14, 163712, 277688, 911008, 1080328, 1653520, 1935128 and 4586240 are such terms and 2^2677*(2680*2^2677-1) is the largest such term that I have found. (End)

References

• Inspired by David Wells, Curious and Interesting Numbers (Revised), Penguin Books.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..89 (terms < 2*10^12)

Crossrefs

Cf. A000005, A000010, A000203, A079536.

Programs

Mathematica

Do[If[Mod[EulerPhi[n]*DivisorSigma[0, n]-DivisorSigma[1, n], n]==0, Print[n]], {n, 1, 1.05*10^7}]
Select[Range[6000000],Divisible[EulerPhi[#]DivisorSigma[0,#]- DivisorSigma[ 1,#], #]&] (* Harvey P. Dale, Mar 10 2012 *)

PARI

isok(k) = {my(f=factor(k)); (eulerphi(f)*numdiv(f)-sigma(f))%k == 0; } \\ Jinyuan Wang, Mar 17 2020

Extensions

More terms from Jinyuan Wang, Mar 17 2020