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

1,1

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

Cf. A000203, A000396, A002025, A002046, A017665, A017666, A253535.

Sequence in context: A228181 A196208 A196475 * A255136 A087252 A141274

Adjacent sequences: A253531 A253532 A253533 * A253535 A253536 A253537

Keyword

nonn

Author

Michel Marcus, Jan 03 2015

Status

approved