

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).


2



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
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OFFSET

1,2


COMMENTS

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) = p1 = 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.


LINKS

Table of n, a(n) for n=1..60.


EXAMPLE

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.


MAPLE

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


MATHEMATICA

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


PROG

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


CROSSREFS

Similar for: A011257 (phi*sigma square), A327830 (sigma*tau square).
Subsequences: A020488, A090698.
Cf. A341939, A341940.
Cf. A000005 (tau), A000010 (phi).
Sequence in context: A104816 A181022 A185870 * A341939 A244353 A143144
Adjacent sequences: A341935 A341936 A341937 * A341939 A341940 A341941


KEYWORD

nonn


AUTHOR

Bernard Schott, Feb 24 2021


STATUS

approved



