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A020492 Balanced numbers: numbers n such that phi(n) (A000010) divides sigma(n) (A000203). 51
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; text; internal format)
OFFSET

1,2

COMMENTS

The quotient A020492(n)/A002088(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 or zeta(2)^2 [~2.705808084277845]. - Labos Elemer, Sep 20 2004, corrected by Charles R Greathouse IV, Jun 20 2012

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, Nov 28 2005

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. - Enrique Pérez Herrero, Sep 05 2010

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.

LINKS

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

EXAMPLE

sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.

MAPLE

with(numtheory);

A020492:=proc(q)

local n; for n from 1 to q do if (sigma(n) mod phi(n))=0 then print(n);

fi; od; end:

A020492(10000000); # Paolo P. Lava, Jan 31 2013.

MATHEMATICA

Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]

PROG

(MAGMA) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; [From Klaus Brockhaus, Nov 09 2008]

(PARI) select(n->sigma(n)%eulerphi(n)==0, vector(10^4, i, i)) \\ Charles R Greathouse IV, Jun 20 2012

CROSSREFS

Cf. A000010, A000203, A000043, A000668, A011257.

Sequence in context: A015769 A015765 A015771 * A110590 A281110 A252792

Adjacent sequences:  A020489 A020490 A020491 * A020493 A020494 A020495

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

More terms from Farideh Firoozbakht, Nov 28 2005

STATUS

approved

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Last modified April 27 06:52 EDT 2017. Contains 285508 sequences.