A340816 Numbers k for which sigma(k^2)/k^2 reaches a new record, where sigma = A000203.

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 2520, 4620, 9240, 13860, 27720, 55440, 60060, 120120, 180180, 360360, 720720, 1441440, 1801800, 2042040, 3063060, 6126120, 12252240, 24504480, 30630600, 36756720, 38798760, 58198140, 116396280

Offset: 1

Author

Robert Israel, Jan 22 2021

Keywords

• nonn

Comments

Appears to be almost identical to A308471.

Examples

			a(1) = 1 with sigma(1^2)/1^2 = 1.
a(2) = 2 with sigma(2^2)/2^2 = 7/4 > 1.
3 is not in the sequence because sigma(3^2)/3^2 = 13/9 < 7/4.
a(3) = 4 with sigma(4^2)/4^2 = 31/16 > 7/4.
		

Links

• MathOverflow, On superabundant-like numbers

Crossrefs

Cf. A000203, A004394, A308471.

Programs

Maple

wmax:= 0: R:= NULL:
for n from 1 to 10^6 do
  w:= numtheory:-sigma(n^2)/n^2;
  if w > wmax then
    wmax:= w; R:= R, n;
  fi;
od:
R;

Mathematica

DeleteDuplicates[Table[{k,DivisorSigma[1,k^2]/k^2},{k,31*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 30 terms of the sequence. To generate more increase the k constant (now set at 31*10^5) but the program may take a long time to run. *) (* Harvey P. Dale, Sep 02 2023 *)