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

1,2

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.

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[#]] &]

Prog

(Magma) IsInteger := func<i | i eq Floor(i)>; [n: n in [1..300] | IsInteger((n*NumberOfDivisors(n)+DivisorSigma(2, n))/(2*SumOfDivisors(n)))];
(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
(PARI) isok(n) = denominator((n*sigma(n, 0) + sigma(n, 2))/(2*sigma(n)))==1; \\ Michel Marcus, Jan 14 2016

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.

Sequence in context: A067019 A132106 A242393 * A338483 A318871 A330225

Adjacent sequences: A210491 A210492 A210493 * A210495 A210496 A210497

Keyword

nonn

Author

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

Status

approved