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

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A000043, A000668.

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

Programs

Magma

[k:k in [1..4500]| IsPower(EulerPhi(k)*DivisorSigma(1,k),2)]; // Marius A. Burtea, Sep 19 2019

Mathematica

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

PARI

is(n)=issquare(eulerphi(n)*sigma(n)) \\ Charles R Greathouse IV, May 09 2013