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

1,1

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

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

Crossrefs

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

Sequence in context: A335163 A365963 A095867 * A349171 A338658 A344224

Adjacent sequences: A253532 A253533 A253534 * A253536 A253537 A253538

Keyword

nonn

Author

Michel Marcus, Jan 03 2015

Status

approved