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A342358
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Balanced numbers (A020492) that are also arithmetic numbers (A003601) and harmonic numbers (A001599).
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0
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OFFSET
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1,2
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COMMENTS
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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 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.
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
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LINKS
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Table of n, a(n) for n=1..8.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.
Oystein Ore, On the averages of the divisors of a number (annotated scanned copy).
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EXAMPLE
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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.
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MAPLE
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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]);
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MATHEMATICA
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Select[Range[350000], And @@ Divisible[(s = DivisorSigma[1, #]), {(d = DivisorSigma[0, #]), EulerPhi[#]}] && Divisible[#*d, s] &] (* Amiram Eldar, Mar 09 2021 *)
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PROG
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(PARI) isok(m) = my(s=sigma(m), t=numdiv(m)); !(s % eulerphi(m)) && !(s % t) && !((m*t) % s); \\ Michel Marcus, Mar 09 2021
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CROSSREFS
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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
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KEYWORD
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nonn,more
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AUTHOR
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Bernard Schott, Mar 09 2021
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EXTENSIONS
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a(6)-a(8) from Amiram Eldar, Mar 09 2021
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STATUS
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approved
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