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%I #19 May 09 2021 02:27:12
%S 12,28,30,40,44,56,84,96,117,120,135,140,182,184,190,198,224,234,248,
%T 252,260,264,270,280,284,308,318,360,380,420,462,476,496,496,546,564,
%U 570,580,585,585,618,630,672,752,812,819,840,855,910,924,946,992
%N Larger member of a harmonious pair.
%C 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.
%C An amicable pair forms a harmonious pair, so the larger member of an amicable pair A002046 is a term of this sequence.
%C An integer can form a harmonious pair with several lesser integers; the first example is (496,28) and (496,6).
%C 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
%H Amiram Eldar, <a href="/A253534/b253534.txt">Table of n, a(n) for n = 1..1000</a>
%H Jamie Bishop, Abigail Bozarth, Rebekah Kuss, and Benjamin Peet, <a href="https://arxiv.org/abs/2104.11366">The Abundancy Index and Feebly Amicable Numbers</a>, arXiv:2104.11366 [math.NT], 2021.
%H M. Kozek, F. Luca, P. Pollack, and C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/KozekLucaPollackPomeranceIJNTv4.pdf">Harmonious numbers</a>, IJNT, to appear.
%e 4 and 12 form a harmonious pair since 4/sigma(4) + 12/sigma(12) = 4/7 + 3/7 = 1.
%t 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 *)
%o (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);}
%o lista(nn) = {for (n=1, nn, for (i=1, nbsh(n), print1(n, ", ")););}
%Y Cf. A000203, A000396, A002025, A002046, A017665, A017666, A253535.
%K nonn
%O 1,1
%A _Michel Marcus_, Jan 03 2015
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