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A210494 Biharmonic numbers: numbers m such that ( Hd(m)+Cd(m) )/2 is an integer, where Hd(m) and Cd(m) are the harmonic mean and the contraharmonic (or antiharmonic) mean of the divisors of m.

%I A210494 #71 Sep 08 2022 08:46:01
%S A210494 1,3,5,7,11,13,17,19,23,29,31,35,37,41,43,47,53,59,61,67,71,73,79,83,
%T A210494 89,97,101,103,107,109,113,119,127,131,137,139,149,151,157,163,167,
%U A210494 173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263
%N A210494 Biharmonic numbers: numbers m such that ( Hd(m)+Cd(m) )/2 is an integer, where Hd(m) and Cd(m) are the harmonic mean and the contraharmonic (or antiharmonic) mean of the divisors of m.
%C A210494 Equivalently, numbers m such that ( m*sigma_0(m)+sigma_2(m) ) / (2*sigma_1(m)) = (A038040(m) + A001157(m))/A074400(m) is an integer.
%C A210494 All odd primes belong to the sequence. In fact, if p is an odd prime, (p*sigma_0(p)+sigma_2(p))/(2*sigma_1(p)) = (p+1)/2, therefore p is a biharmonic number.
%H A210494 Amiram Eldar, <a href="/A210494/b210494.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Bruno Berselli)
%H A210494 Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="http://arxiv.org/abs/1601.03081">The Biharmonic mean</a>, arXiv:1601.03081 [math.NT], 2016, pages 6-14.
%H A210494 Umberto Cerruti, <a href="/A210494/a210494.pdf">Numeri Armonici e Numeri Perfetti</a> (in Italian), 2013. The sequence is on page 13.
%p A210494 with(numtheory); P:=proc(q) local a,k,n;
%p A210494 for n from 1 to q do a:=divisors(n);
%p A210494 if type((n*tau(n)+add(a[k]^2,k=1..nops(a)))/(2*sigma(n)),integer) then print(n); fi; od; end; P(1000); # _Paolo P. Lava_, Oct 11 2013
%t A210494 B[n_] := (n DivisorSigma[0, n] + DivisorSigma[2, n])/(2 DivisorSigma[1, n]); Select[Range[300], IntegerQ[B[#]] &]
%o A210494 (Magma) IsInteger := func<i | i eq Floor(i)>; [n: n in [1..300] | IsInteger((n*NumberOfDivisors(n)+DivisorSigma(2,n))/(2*SumOfDivisors(n)))];
%o A210494 (Haskell)
%o A210494 a210494 n = a210494_list !! (n-1)
%o A210494 a210494_list = filter
%o A210494    (\x -> (a001157 x + a038040 x) `mod` a074400 x == 0) [1..]
%o A210494 -- _Reinhard Zumkeller_, Jan 21 2014
%o A210494 (PARI) isok(n) = denominator((n*sigma(n,0) + sigma(n,2))/(2*sigma(n)))==1; \\ _Michel Marcus_, Jan 14 2016
%Y A210494 Cf. A001599 (harmonic numbers), A020487 (antiharmonic numbers), A038040 (n*sigma_0(n)), A001157 (sigma_2(n)), A074400 (2*sigma_1(n)), A230214 (nonprime terms of A210494).
%Y A210494 Cf. A189835.
%K A210494 nonn
%O A210494 1,2
%A A210494 _Bruno Berselli_, Oct 03 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)