A253535 Lesser member of a harmonious pair.

4, 6, 14, 10, 20, 8, 15, 26, 60, 2, 42, 14, 66, 88, 102, 45, 10, 4, 174, 153, 164, 38, 15, 22, 220, 182, 110, 9, 92, 33, 345, 190, 6, 28, 285, 195, 435, 68, 78, 364, 315, 207, 2, 368, 248, 42, 51, 846, 790, 21, 870, 32, 334, 558, 82, 34, 117, 1184, 598, 574

Offset: 1

Author

Michel Marcus, Jan 03 2015

Keywords

• nonn

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

Examples

			4 and 12 form a harmonious pair since 4/sigma(4) + 12/sigma(12) = 4/7 + 3/7 = 1.
		

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.

Crossrefs

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

Programs

Mathematica

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 *)

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, #nbshn, print1(nbshn[i], ", ")););}