

A342358


Balanced numbers (A020492) that are also arithmetic numbers (A003601) and harmonic numbers (A001599).


0




OFFSET

1,2


COMMENTS

Equivalently, numbers m such that sigma(m)/phi(m), sigma(m)/tau(m) and m*tau(m)/sigma(m) are all integers where phi = A000010, tau = A000005 and sigma = A000203.
Conjecture: 1 would be the only odd term of this sequence, because Oystein Ore conjectured that 1 is the only odd harmonic number (see link), and 1 is an arithmetic and balanced number (A342103).
Proposition: there are no primes in the sequence. Proof: the only prime that is both arithmetic and balanced is 3 (A342103), but 3 is not an harmonic number.
As HansJoachim Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0 (see link), the asymptotic density of this sequence is also 0.
a(9) > 6.5*10^14 (verified using list of balanced numbers from Jud McCranie). All the numbers in this range that are both balanced and harmonic numbers are also arithmetic numbers.  Amiram Eldar, Mar 09 2021


LINKS

Table of n, a(n) for n=1..8.
HansJoachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371374.
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615619.
Oystein Ore, On the averages of the divisors of a number (annotated scanned copy).


EXAMPLE

For 6: tau(6) = 4, phi(6) = 2, sigma(6) = 12, 6*tau(6)/sigma(6) = 6*4/12 = 2, sigma(6)/tau(6) = 3 and sigma(6)/phi(6) = 2, hence 6 is a term.


MAPLE

with(numtheory): filter:= q > (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) = 0 and (q*tau(q) mod sigma(q) = 0) : select(filter, [$1..300000]);


MATHEMATICA

Select[Range[350000], And @@ Divisible[(s = DivisorSigma[1, #]), {(d = DivisorSigma[0, #]), EulerPhi[#]}] && Divisible[#*d, s] &] (* Amiram Eldar, Mar 09 2021 *)


PROG

(PARI) isok(m) = my(s=sigma(m), t=numdiv(m)); !(s % eulerphi(m)) && !(s % t) && !((m*t) % s); \\ Michel Marcus, Mar 09 2021


CROSSREFS

Intersection of A001599, A003601 and A020492.
Intersection of A001599 and A342103.
Intersection of A007340 and A020492.
Cf. A000005, A000010, A000203.
Sequence in context: A090944 A007340 A335318 * A122483 A335369 A335388
Adjacent sequences: A342355 A342356 A342357 * A342359 A342360 A342361


KEYWORD

nonn,more


AUTHOR

Bernard Schott, Mar 09 2021


EXTENSIONS

a(6)a(8) from Amiram Eldar, Mar 09 2021


STATUS

approved



