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 A011257 Geometric mean of phi(n) and sigma(n) is an integer. 21
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For these terms the arithmetic mean is also an integer. It is conjectured that sigma(n) 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 (phi(m)*sigma(m))^(1/2)=2^(p-1)*(2^(p-1)-1) is an integer. So for n>1, 2^(A000043(n)-2)*2^(A000043(n)-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 M. F. Hasler and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from 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, 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 STATUS approved

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Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)