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%I #32 Mar 11 2021 20:46:05
%S 1,6,140,270,2970,332640,14303520,5297292000
%N Balanced numbers (A020492) that are also arithmetic numbers (A003601) and harmonic numbers (A001599).
%C 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.
%C 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).
%C 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.
%C As Hans-Joachim Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0 (see link), the asymptotic density of this sequence is also 0.
%C 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
%H Hans-Joachim Kanold, <a href="http://dx.doi.org/10.1007/BF01342887">Über das harmonische Mittel der Teiler einer natürlichen Zahl</a>, Math. Ann., Vol. 133 (1957), pp. 371-374.
%H Oystein Ore, <a href="http://www.jstor.org/stable/2305616">On the averages of the divisors of a number</a>, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.
%H Oystein Ore, <a href="/A001599/a001599.pdf">On the averages of the divisors of a number</a> (annotated scanned copy).
%e 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.
%p 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]);
%t Select[Range[350000], And @@ Divisible[(s = DivisorSigma[1, #]), {(d = DivisorSigma[0, #]), EulerPhi[#]}] && Divisible[#*d, s] &] (* _Amiram Eldar_, Mar 09 2021 *)
%o (PARI) isok(m) = my(s=sigma(m), t=numdiv(m)); !(s % eulerphi(m)) && !(s % t) && !((m*t) % s); \\ _Michel Marcus_, Mar 09 2021
%Y Intersection of A001599, A003601 and A020492.
%Y Intersection of A001599 and A342103.
%Y Intersection of A007340 and A020492.
%Y Cf. A000005, A000010, A000203.
%K nonn,more
%O 1,2
%A _Bernard Schott_, Mar 09 2021
%E a(6)-a(8) from _Amiram Eldar_, Mar 09 2021
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