

A011257


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


23



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
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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^(p2)*(2^p1) is in the sequence because phi(m) = 2^(p2)*(2^(p1)1); sigma(m) = (2^(p1)1)*2^p hence sqrt(phi(m)*sigma(m)) = 2^(p1)*(2^(p1)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 BroughanFordLuca 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 MingZhi (typescript submitted to Unsolved Problems section of Monthly, 960110).


LINKS



MATHEMATICA

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


PROG

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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



