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A011257 Numbers k such that the geometric mean of phi(k) and sigma(k) is an integer. 23

%I #47 Nov 03 2023 22:36:42

%S 1,14,30,51,105,170,194,248,264,364,405,418,477,595,679,714,760,780,

%T 1023,1455,1463,1485,1496,1512,1524,1674,1715,1731,1796,1804,2058,

%U 2080,2651,2754,2945,3080,3135,3192,3410,3534,3567,3596,3828,3956,4064,4381,4420

%N Numbers k such that the geometric mean of phi(k) and sigma(k) is an integer.

%C For these terms the arithmetic mean is also an integer. It is conjectured that sigma(k) for these numbers is never odd. See also A065146, A028982, A028983. - _Labos Elemer_, Oct 18 2001

%C If p > 2 and 2^p - 1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sqrt(phi(m)*sigma(m)) = 2^(p-1)*(2^(p-1)-1) is an integer. So for j > 1, 2^(A000043(j)-2)*2^(A000043(j)-1) is in the sequence. - _Farideh Firoozbakht_, Nov 27 2005

%C From a(2633) = 6931232 on, it is no longer true (as was once conjectured) that a(n) > n^2. - _M. F. Hasler_, Feb 07 2009

%C It follows from Theorems 1 and 2 in Broughan-Ford-Luca that a(n) << n^(24+e) for all e > 0. - _Charles R Greathouse IV_, May 09 2013

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.

%D Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, 96-01-10).

%H Charles R Greathouse IV, <a href="/A011257/b011257.txt">Table of n, a(n) for n = 1..10000</a> (first 2000 terms from M. F. Hasler)

%H K. Broughan, K. Ford, and F. Luca, <a href="http://www.math.waikato.ac.nz/~kab/papers/BroughanFordLucaCollMath.pdf">On square values of the product of the Euler totient function and sum of divisors function</a>, Colloquium Mathematicum, (to appear).

%H Tristan Freiberg, <a href="http://arxiv.org/abs/1008.1978">Products of shifted primes simultaneously taking perfect power values</a>, Journal of the Australian Mathematical Society 92:2 (2012), pp. 145-154. arXiv:1008.1978 [math.NT], 2010.

%H Richard K. Guy, <a href="http://www.jstor.org/stable/2974586">Divisors and desires</a>, Amer. Math. Monthly, 104 (1997), 359-360.

%H Luis Elesban Santos Cruz and Florian Luca, <a href="http://dx.doi.org/10.2140/involve.2015.8.745">Power values of the product of the Euler function and the sum of divisors function</a>, involve, Vol. 8 (2015), No. 5, 745-748.

%t Select[Range[8000], IntegerQ[Sqrt[DivisorSigma[1, #] EulerPhi[#]]] &] (* _Carl Najafi_, Aug 16 2011 *)

%o (PARI) is(n)=issquare(eulerphi(n)*sigma(n)) \\ _Charles R Greathouse IV_, May 09 2013

%o (Magma) [k:k in [1..4500]| IsPower(EulerPhi(k)*DivisorSigma(1,k),2)]; // _Marius A. Burtea_, Sep 19 2019

%Y Cf. A000043, A000668.

%Y Cf. A293391 (sigma(m)/phi(m) is a perfect square), A327624 (this sequence \ A293391).

%K nonn

%O 1,2

%A _N. J. A. Sloane_

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