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A020492
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Balanced numbers: numbers n such that phi(n) (A000010) divides sigma(n) (A000203).
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49
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1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The quotient A020492[n]/A002088[n] = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^2/36 or Zeta(2))^2 [~2.705808084277845]. - Labos E. (labos(AT)ana.sote.hu), Sep 20 2004
If 2^p-1 is prime (a Mersenne prime) then m=2^(p-2)*(2^p-1) is in the sequence because when p=2 we get m=3 and phi(3) divides sigma(3) and for p>2, phi(m)=2^(p-2)*(2^(p-1)-1); sigma(m) =(2^(p-1)-1)*2^p hence sigma(m)/phi(m)=4 is an integer. So for each n, 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht (mymontain(AT)yahoo.com), Nov 28 2005
Contribution from Enrique Perez Herrero (psychgeometry(AT)gmail.com), Sep 05 2010: (Start)
phi and sigma are both multiplicative functions and for this reason if m and n are coprime and included in this sequence then:
m*n is also in this sequence. (End)
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REFERENCES
| D. Chiang, "N's for which phi(N) divides sigma(N)", Mathematical Buds, Chap. VI pp. 53-70 Vol. 3 Ed. H. D. Ruderman, Mu Alpha Theta 1984.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
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EXAMPLE
| sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.
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MATHEMATICA
| Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]
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PROG
| (MAGMA) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 09 2008]
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CROSSREFS
| Cf. A000010, A000203.
Cf. A000043, A000668, A011257.
Sequence in context: A015769 A015765 A015771 * A110590 A111271 A070926
Adjacent sequences: A020489 A020490 A020491 * A020493 A020494 A020495
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| More terms from Farideh Firoozbakht (mymontain(AT)yahoo.com), Nov 28 2005
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