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

1, 14, 30, 51, 105, 170, 194, 248, 264, 364, 405, 418, 477, 595, 679, 714, 760, 780, 1023, 1455, 1463, 1485, 1496, 1512, 1524, 1674, 1715, 1731, 1796, 1804, 2058, 2080, 2651, 2754, 2945, 3080, 3135, 3192, 3410, 3534, 3567, 3596, 3828, 3956, 4064, 4381, 4420

Offset

1,2

Comments

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

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

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

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

References

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

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from M. F. Hasler)

K. Broughan, K. Ford, and F. Luca, On square values of the product of the Euler totient function and sum of divisors function, Colloquium Mathematicum, (to appear).

Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, Journal of the Australian Mathematical Society 92:2 (2012), pp. 145-154. arXiv:1008.1978 [math.NT], 2010.

Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Luis Elesban Santos Cruz and Florian Luca, Power values of the product of the Euler function and the sum of divisors function, involve, Vol. 8 (2015), No. 5, 745-748.

Mathematica

Select[Range[8000], IntegerQ[Sqrt[DivisorSigma[1, #] EulerPhi[#]]] &] (* Carl Najafi, Aug 16 2011 *)

Prog

(PARI) is(n)=issquare(eulerphi(n)*sigma(n)) \\ Charles R Greathouse IV, May 09 2013
(Magma) [k:k in [1..4500]| IsPower(EulerPhi(k)*DivisorSigma(1, k), 2)]; // Marius A. Burtea, Sep 19 2019

Crossrefs

Cf. A000043, A000668.

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

Sequence in context: A044075 A044456 A132759 * A083540 A027575 A104776

Adjacent sequences: A011254 A011255 A011256 * A011258 A011259 A011260

Keyword

nonn

Author

N. J. A. Sloane

Status

approved