A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 232, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset

1,2

Comments

The first 20 terms of this sequence are also the first 20 terms of A144695: m such that sigma(m)/tau(m) is a square. Indeed, if sigma(m)/tau(m) is a square then sigma(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A327831; the first one is a(21) = 232.

The primes p of the form 2*k^2 - 1: 7, 17, 31, 71, ... (A066436) form a subsequence because sigma(p) * tau(p) = (2*k)^2.

Another subsequence consists of the terms m such that sigma(m) and tau(m) are both squares; this occurs when m is the product of two distinct primes p*q, p < q where sigma(m) = (p+1)*(q+1) is a square and tau(m) = 4. The first few terms are 22, 94, 115, 119, 214, ... They are in A256152.

Links

Table of n, a(n) for n=1..55.

Example

sigma(30) = 72 and tau(30) = 8, sigma(30)*tau(30) = 576 = 24^2, hence 30 is a term.

Maple

filter:= s -> issqr(sigma(s)*tau(s)) : select(filter, [$1..2500]);

Mathematica

Select[Range[1000], IntegerQ @ Sqrt[DivisorSigma[0, #] * DivisorSigma[1, #]] &] (* Amiram Eldar, Sep 27 2019 *)

Prog

(Magma) [k:k in [1..1150]| IsSquare(#Divisors(k)*DivisorSigma(1, k))]; // Marius A. Burtea, Sep 27 2019
(PARI) isok(m) = issquare(numdiv(m)*sigma(m)); \\ Michel Marcus, Sep 27 2019

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A064840 (tau*sigma).

Cf. A011257 (similar, with phi(m) and sigma(m)), A144695 (sigma(m)/tau(m) is a square), A327831 (sigma(m) * tau(m) is a square but sigma(m)/tau(m) is not an integer).

Subsequences: A066436, A256152.

Sequence in context: A287182 A104480 A053746 * A144695 A125244 A070416

Adjacent sequences: A327827 A327828 A327829 * A327831 A327832 A327833

Keyword

nonn

Author

Bernard Schott, Sep 27 2019

Status

approved