A253535 - OEIS

%I #16 May 09 2021 02:26:57

%S 4,6,14,10,20,8,15,26,60,2,42,14,66,88,102,45,10,4,174,153,164,38,15,

%T 22,220,182,110,9,92,33,345,190,6,28,285,195,435,68,78,364,315,207,2,

%U 368,248,42,51,846,790,21,870,32,334,558,82,34,117,1184,598,574

%N Lesser 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 lesser member of an amicable pair A002025 is a term of this sequence.

%H Amiram Eldar, <a href="/A253535/b253535.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, m]], {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, #nbshn, print1(nbshn[i], ", ")););}

%Y Cf. A000203, A002025, A002046, A017665, A017666, A253534.

%K nonn

%O 1,1

%A _Michel Marcus_, Jan 03 2015