A253534 Larger member of a harmonious pair.

12, 28, 30, 40, 44, 56, 84, 96, 117, 120, 135, 140, 182, 184, 190, 198, 224, 234, 248, 252, 260, 264, 270, 280, 284, 308, 318, 360, 380, 420, 462, 476, 496, 496, 546, 564, 570, 580, 585, 585, 618, 630, 672, 752, 812, 819, 840, 855, 910, 924, 946, 992

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 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

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, A000396, A002025, A002046, A017665, A017666, A253535.

Programs

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 *)

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, ", ")););}