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Search: id:A011257
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| A011257 |
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Geometric mean of phi(n) and sigma(n) is an integer. |
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+0
17
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| 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
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1,2
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COMMENT
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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 E. (labos(AT)ana.sote.hu), 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 (mymontain(AT)yahoo.com), 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
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
R. K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, 96-01-10)
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LINKS
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M. F. Hasler, Table of n, a(n) for n=1,...,2000
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CROSSREFS
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Cf. A000043, A000668.
Sequence in context: A044075 A044456 A132759 this_sequence A083540 A027575 A104776
Adjacent sequences: A011254 A011255 A011256 this_sequence A011258 A011259 A011260
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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