A158299 Numerators of averages of squares of the divisors of n.

1, 5, 5, 7, 13, 25, 25, 85, 91, 65, 61, 35, 85, 125, 65, 341, 145, 455, 181, 91, 125, 305, 265, 425, 217, 425, 205, 175, 421, 325, 481, 455, 305, 725, 325, 637, 685, 905, 425, 1105, 841, 625, 925, 427, 1183, 1325, 1105, 341, 817, 1085, 725, 595, 1405, 1025

Offset: 1

Author

Jaroslav Krizek, Mar 15 2009

Keywords

• nonn
• frac
• changed

Comments

Because Mathematica represents rational numbers with the smallest possible denominator, the terms of the sequence are numerators appropriate to such denominators. For example, the divisors of 3 are 1 and 3, so their squares are 1 and 9. The mean of those squares could be represented as 10/2 or 5/1. Mathematica selects the latter so a(3) is 5 rather than 10. - Harvey P. Dale, Oct 13 2011

If m and n are coprime, f(m*n) divides f(m)*f(n). - Robert Israel, Jul 15 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Cf. A001157, A000005, A158298 (for denominators).

Programs

Maple

f:= proc(n) local D;
  D:= map(t -> t^2,numtheory:-divisors(n));
  numer(convert(D,`+`)/nops(D));
end proc:
map(f, [$1..100]); # Robert Israel, Jul 15 2019

Mathematica

Numerator[Mean/@(Divisors[Range[60]]^2)] (* Harvey P. Dale, Oct 13 2011 *)
Array[Numerator[DivisorSigma[2, #]/DivisorSigma[0, #]] &, 100]; (* Amiram Eldar, Jul 15 2019 *)

Extensions

Corrected and extended by Harvey P. Dale, Oct 13 2011