A324059 Numbers n such that sigma(n)/(phi(n) + tau(n)) is a record.

1, 2, 4, 6, 10, 12, 18, 24, 30, 42, 60, 84, 90, 120, 180, 210, 360, 420, 840, 1260, 1680, 2520, 4620, 7560, 9240, 13860, 18480, 27720, 55440, 110880, 120120, 180180, 240240, 360360, 720720, 1441440, 2162160, 3603600, 4084080, 4324320, 6126120, 12252240, 24504480

Offset: 1

Author

Robert G. Wilson v, Feb 13 2019

Keywords

• nonn

Comments

sigma(a(69))/(phi(a(69)) + tau(a(69))) = 857304000/23950081 ~= 35.7955.

Number of terms =< 10^k, k=0,1,2,3: 1, 5, 13, 19, 25, 29, 35, 41, 46, 50, 56, 63, 69, ..., .

All terms so far except 10, 18, 42, 84, 90 are in A025487. - David A. Corneth, Feb 14 2019

Examples

			a(7) = 18 since it is the first number greater than a(6) such that sigma(18)/(phi(18) + tau(18)) = 13/4 > 14/5 =  sigma(12)/(phi(12) + tau(12)).
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..70

Crossrefs

Cf. A000010, A000005, A000203, A061468, A324060, A025487.

Programs

Maple

Res:= NULL: mx:= 0: count:= 0:
for n from 1 while count < 60 do
  v:= numtheory:-sigma(n)/(numtheory:-phi(n)+numtheory:-tau(n));
  if v > mx then
    mx:= v;
    count:= count+1;
    Res:= Res, n;
  fi
od:
Res; # Robert Israel, Feb 13 2019

Mathematica

k = 1; mx = 0; lst = {}; While[k < 25000000, If[ DivisorSigma[1, k] > mx (EulerPhi[k] + DivisorSigma[0, k]), mx = DivisorSigma[1, k]/(EulerPhi[k] + DivisorSigma[0, k]); AppendTo[lst, k]]; k ++]; lst
DeleteDuplicates[Table[{n,DivisorSigma[1,n]/(EulerPhi[n]+DivisorSigma[0,n])},{n,2451*10^4}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, Jun 08 2022 *)

PARI

lista(nn) = {my(m=0, newm); for (n=1, nn, newm = sigma(n)/(eulerphi(n) + numdiv(n)); if (newm > m, print1(n, ", "); m = newm););} \\ Michel Marcus, Feb 13 2019