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.

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Author

Bruno Berselli, Oct 03 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

Keywords

• nonn

Comments

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.

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Bruno Berselli)
• Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016, pages 6-14.
• Umberto Cerruti, Numeri Armonici e Numeri Perfetti (in Italian), 2013. The sequence is on page 13.

Crossrefs

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).

Cf. A189835.

Programs

Haskell

a210494 n = a210494_list !! (n-1)
a210494_list = filter
   (\x -> (a001157 x + a038040 x) `mod` a074400 x == 0) [1..]
-- Reinhard Zumkeller, Jan 21 2014

Magma

IsInteger := func; [n: n in [1..300] | IsInteger((n*NumberOfDivisors(n)+DivisorSigma(2,n))/(2*SumOfDivisors(n)))];

Maple

with(numtheory); P:=proc(q) local a,k,n;
for n from 1 to q do a:=divisors(n);
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

Mathematica

B[n_] := (n DivisorSigma[0, n] + DivisorSigma[2, n])/(2 DivisorSigma[1, n]); Select[Range[300], IntegerQ[B[#]] &]

PARI

isok(n) = denominator((n*sigma(n,0) + sigma(n,2))/(2*sigma(n)))==1; \\ Michel Marcus, Jan 14 2016