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

1,2

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. [From 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)

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 *)

Crossrefs

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

Sequence in context: A078551 A247877 A252006 * A093307 A264388 A141392

Adjacent sequences: A158296 A158297 A158298 * A158300 A158301 A158302

Keyword

nonn,frac

Author

Jaroslav Krizek, Mar 15 2009

Extensions

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

Status

approved