A065146 -id:A065146 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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).

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.

Cf. A000043, A000668.

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

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

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

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

A065150 Numbers k such that the harmonic mean of phi(k) and sigma(k) is an integer.

Original entry on oeis.org

1, 12, 15, 35, 56, 78, 95, 140, 143, 172, 190, 248, 264, 287, 315, 319, 323, 357, 418, 477, 588, 594, 675, 812, 814, 840, 899, 910, 1045, 1107, 1118, 1131, 1199, 1208, 1254, 1349, 1420, 1425, 1485, 1495, 1558, 1608, 1672, 1763, 2214, 2261, 2318, 2337

Offset: 1

Labos Elemer, Oct 18 2001

nonn

			m = 319, phi(319) = 280, sigma(319) = 360; phi(319)*sigma(319) = 100800, phi(319) + sigma(319) = 640; 1/(harmonic mean) = (640/100800)/2, harmonic mean = 315, arithmetic mean = 320, geometric mean is not an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010, A000203, A065146, A011257.

Select[Range[2400], IntegerQ[HarmonicMean @ {EulerPhi[#], DivisorSigma[1, #]}] &] (* Amiram Eldar, Mar 20 2025 *)

{ n=0; for (m=1, 10^9, e=eulerphi(m); s=sigma(m); h=(2*e*s)/(e + s); if (frac(h) == 0, write("b065150.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 13 2009

G^2 mod A = 0, where G^2 = A000010(m)*A000203(m), A = (A000010(m) + A000203(m))/2; harmonic mean = (G^2)/A is an integer.

A291959 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and psi(n) are all integers, where phi(n) is the Euler totient function (A000010) and psi(n) is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 150, 300, 418, 450, 525, 600, 750, 836, 900, 1200, 1350, 1500, 1575, 1672, 1800, 2250, 2400, 2625, 2700, 3000, 3135, 3344, 3600, 3675, 3750, 4050, 4500, 4598, 4725, 4800, 5400, 6000, 6688, 6750, 7200, 7500, 7875, 7942, 8100, 9000, 9196, 9405, 9600, 10800

Offset: 1

Amiram Eldar, Sep 06 2017

nonn

The number of terms below 10^3, 10^4, ... are 10, 44, 147, 397, ...

			phi(150)=40, psi(150)=360, their arithmetic mean = 200, geometric mean = 120, harmonic mean = 72 are all integers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A001615, A065146.

dedekindPsi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]];
aQ[n_] := Module[{s = dedekindPsi[n], p = EulerPhi[n]}, IntegerQ[(s + p)/2] && IntegerQ[Sqrt[s*p]] && IntegerQ[2 s*p/(s + p)]]; Select[Range[10^5], aQ]

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

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A065150 Numbers k such that the harmonic mean of phi(k) and sigma(k) is an integer.

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A291959 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and psi(n) are all integers, where phi(n) is the Euler totient function (A000010) and psi(n) is the Dedekind psi function (A001615).

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