OFFSET
1,1
COMMENTS
Let sigma be the usual sum-of-divisors function. We say that x and y form a harmonious pair if x/sigma(x) + y/sigma(y) = 1. Equivalently, the harmonic mean of sigma(x)/x and sigma(y)/y is 2.
An amicable pair forms a harmonious pair, so the larger member of an amicable pair A002046 is a term of this sequence.
An integer can form a harmonious pair with several lesser integers; the first example is (496,28) and (496,6).
Terms that appear more than once: 496, 585, 1485, 1550, 1892, 2678, 2882, 3472, 4455, 8128, ... The k-th perfect number, A000396(k), appears k times. The first non-perfect number that appears k times for k = 1, 2, 3, ... is 12, 585, 63855, ... - Amiram Eldar, Jun 24 2019
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..1000
Jamie Bishop, Abigail Bozarth, Rebekah Kuss, and Benjamin Peet, The Abundancy Index and Feebly Amicable Numbers, arXiv:2104.11366 [math.NT], 2021.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
EXAMPLE
4 and 12 form a harmonious pair since 4/sigma(4) + 12/sigma(12) = 4/7 + 3/7 = 1.
MATHEMATICA
s={}; Do[r = 1 - n/DivisorSigma[1, n]; Do[If[m/DivisorSigma[1, m] == r, AppendTo[s, n]], {m, 1, n-1}], {n, 1, 1000}]; s (* Amiram Eldar, Jun 24 2019 *)
PROG
(PARI) nbsh(n) = {v = []; vn = n/sigma(n); for (m=1, n-1, if (m/sigma(m) + vn == 1, v = concat(v, m)); ); return (v); }
lista(nn) = {for (n=1, nn, for (i=1, nbsh(n), print1(n, ", ")); ); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Jan 03 2015
STATUS
approved