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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.
3
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
KEYWORD
nonn
AUTHOR
Bruno Berselli, Oct 03 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)
STATUS
approved