A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

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

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, A133028(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

The quotients sigma(n)/phi(n) are in A023897. - Bernard Schott, Jun 06 2017

There are 544768 balanced numbers < 10^14. - Jud McCranie, Sep 10 2017

a(975807) = 419998185095132. - Jud McCranie, Nov 28 2017

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

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

Jud McCranie, 670314 balanced numbers (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)

Example

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

Mathematica

Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]
(* Second program: *)
Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* Michael De Vlieger, Nov 28 2017 *)

Prog

(Magma) [ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // 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
(Python)
from sympy import totient, divisor_sigma
print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # Indranil Ghosh, Jul 06 2017
(Python)
from math import prod
from itertools import count, islice
from sympy import factorint
def A020492_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue, 1)):
        f = factorint(m)
        if not prod(p**(e+2)-p for p, e in f.items())%(m*prod((p-1)**2 for p in f)):
            yield m
A020492_list = list(islice(A020492_gen(), 20)) # Chai Wah Wu, Aug 12 2024

Crossrefs

Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).

Positions of 0's in A063514.

Sequence in context: A015769 A015765 A015771 * A110590 A329401 A291174

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