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A253534 Larger member of a harmonious pair. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 15 16:57 EDT 2022. Contains 356148 sequences. (Running on oeis4.)