A327830 -id:A327830 - cp's OEIS frontend

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

Bernard Schott, Oct 14 2019

nonn

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

			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.

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).

[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

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

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 *)

isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ Michel Marcus, Oct 15 2019

A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 54, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341939: m such that phi(m)/tau(m) is the square of an integer. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is a(12) = 54.
If k and q are terms and coprimes, then k*q is another term.
Some subsequences (see examples):
-> The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so tau(p)*phi(p) = (2*k)^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset with tau(p^3)*phi(p^3) = (2*k*p)^2.

			phi(18) = tau(18) = 6, so phi(18)*tau(18) = 6^2.
phi(19) = 18, tau(19) = 2, so phi(19)*tau(19) = 36 = 6^2.
phi(34) = 16, tau(34) = 4, so phi(34)*tau(34) = 16*4 = 64 = 8^2.
phi(125) = 100, tau(125) = 4, so phi(125)*tau(125) = 400 = 20^2.

Similar for: A011257 (phi*sigma square), A327830 (sigma*tau square).
Subsequences: A020488, A090698.
Cf. A341939, A341940.
Cf. A000005 (tau), A000010 (phi).

with(numtheory): filter:= n -> issqr(phi(n)*tau(n)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ GeometricMean[{DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = issquare(numdiv(m)*eulerphi(m)); \\ Michel Marcus, Feb 24 2021

cp's OEIS Frontend

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

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