A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer.

232, 2152, 3240, 3560, 3944, 6516, 17908, 22504, 23716, 26172, 32360, 34344, 36584, 37736, 43300, 45612, 48204, 55080, 55912, 60520, 61480, 69352, 73084, 78184, 79056, 79300, 96552, 104168, 105832, 106088, 125356, 130432, 133864, 140040, 149992, 163764, 168424, 172840, 176360, 183204

Offset

1,1

Comments

If sigma(m)/tau(m) is a square (m is in A144695) then sigma(m)*tau(m) is also a square (m is in A327830), but the converse is false (see 232 in the Example section). This sequence consists of these counterexamples.

It seems that all terms are even. - Marius A. Burtea, Oct 15 2019

Links

Table of n, a(n) for n=1..40.

Example

sigma(232) = 450 and tau(232) = 8, so sigma(232)*tau(232) = 450*8 = 3600 = 60^2 and sigma(232)/tau(232) = 450/8 = 225/4 is not an integer, hence 232 is a term.

Maple

filter:= u -> sigma(u)/tau(u) <> floor(sigma(u)/tau(u)) and issqr(sigma(u)*tau(u)) : select(filter, [$1..100000]);

Mathematica

sQ[n_] := IntegerQ@Sqrt[n]; aQ[n_] := sQ[(d = DivisorSigma[0, n]) * (s = DivisorSigma[1, n])] && !sQ[s/d]; Select[Range[2*10^5], aQ] (* Amiram Eldar, Oct 15 2019 *)

Prog

(PARI) isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ Michel Marcus, Oct 15 2019
(Magma) [k:k in [1..200000] | not IsIntegral(a/b) and IsSquare(a*b) where a is DivisorSigma(1, k) where b is #Divisors(k)]; // Marius A. Burtea, Oct 15 2019

Crossrefs

Equals A144695 \ A327830.

Similar to A327624 with sigma(m) and phi(m).

Cf. A003601 (sigma(m)/tau(m) is an integer), A023883 (sigma(m)/tau(m) is an integer and m is nonprime).

Cf. A000005 (tau), A000203 (sigma).

Sequence in context: A234683 A366816 A238919 * A234682 A156391 A279660

Adjacent sequences: A327828 A327829 A327830 * A327832 A327833 A327834

Keyword

nonn

Author

Bernard Schott, Oct 14 2019

Status

approved