A318084 Numbers m such that sigma(sigma(m))/m is a square.

1, 15, 50, 100, 168, 1023, 1444, 1470, 1600, 1944, 3179, 3822, 4000, 5120, 5776, 6174, 9025, 10752, 12348, 14440, 15125, 21970, 26250, 28416, 28665, 29127, 37544, 39200, 45630, 47151, 49392, 52500, 60984, 66125, 67200, 69819, 71485, 77175, 80000, 90250, 100254, 102300, 102400

Offset: 1

Author

Michel Marcus, Aug 15 2018

Keywords

• nonn

Comments

This is a necessary condition to have sigma(sigma(m))/sigma(m) = sigma(m)/m.

Are there other integers than 1, for which this is satisfied?

If m is an odd number such that sigma(sigma(m^2))/2 is a square, and p is in A000043 such that 2^p-1 does not divide sigma(m^2), then 2^(p-1)*m^2 is in the sequence. Such m include 5, 19, 161, 543, 1031, 1899, 3035, 6673. Thus if A000043 is infinite, so is this sequence. - Robert Israel, Aug 17 2018

Links

• Giovanni Resta, Table of n, a(n) for n = 1..5000 (first 200 terms from Robert Israel)

Crossrefs

Cf. A000203 (sigma), A000043, A051027, A006532, A069070.

Cf. A318059, A318060, A318083.

Programs

Maple

filter:= proc(n) local t; t:= (numtheory:-sigma @@2)(n)/n; issqr(numer(t)) and issqr(denom(t)) end proc:select(filter, [$1..200000]); # Robert Israel, Aug 17 2018

Mathematica

Select[Range[10^5], IntegerQ@ Sqrt[ DivisorSigma[1, DivisorSigma[1, #]] #] &] (* Giovanni Resta, Aug 19 2018 *)

PARI

isok(n) = issquare(sigma(sigma(n))/n);