A253535 -id:A253535 - cp's OEIS frontend

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

Michel Marcus, Jan 03 2015

nonn

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

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

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.

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

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 *)

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, ", ")););}

A384487 Numbers k such that there exist two integers 0

Original entry on oeis.org

396, 504, 600, 756, 840, 924, 1056, 1080, 1140, 1170, 1260, 1320, 1428, 1440, 1488, 1512, 1540, 1560, 1596, 1638, 1650, 1656, 1680, 1704, 1710, 1740, 1800, 1820, 1840, 1848, 1872, 1932, 1980, 2016, 2040, 2100, 2160, 2184, 2232, 2244, 2256, 2280, 2340, 2352, 2380, 2400, 2430, 2436, 2448, 2460, 2484

Offset: 1

S. I. Dimitrov, Jun 01 2025

nonn

The numbers i, j and k form a WHM(1)-amicable triple (WHM = weighted harmonic mean). See Dimitrov link.

			504 is a term because (72, 360, 504) is a triple with 72/sigma(72) + 360/sigma(360) + 504/sigma(504) = 1.
420 is not a term because the corresponding triple is (84, 420, 420).

Robert Israel, Table of n, a(n) for n = 1..600
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A125490, A125491, A125492, A253534, A253535, A383964.

S:= {}: S2:= {}: R:= NULL: count:= 0:
for k from 1 while count < 100 do
  v:= k/numtheory:-sigma(k);
  if member(1-v,S2) then
    R:= R, k; count:= count+1;
 fi;
  S2:= S2 union map(t -> `if`(t+v<1,t+v,NULL),S);
  S:= S union {v};
od:
R; # Robert Israel, Jul 01 2025

isok(k) = for (i=1, k-1, for (j=i+1, k-1, if (i/sigma(i) + j/sigma(j) + k/sigma(k) == 1, /* print([i,j,k]); */ return(1)););); \\ Michel Marcus, Jun 02 2025

More terms from Michel Marcus, Jun 02 2025

A384706 Integers y such that there exists an integer 0 < x < y such that y/sigma(x) + x/sigma(y) = 1.

Original entry on oeis.org

14, 20, 42, 54, 62, 88, 99, 108, 114, 124, 126, 132, 189, 195, 204, 210, 220, 238, 252, 254, 272, 284, 328, 340, 385, 414, 420, 432, 455, 464, 468, 495, 508, 528, 560, 572, 608, 621, 630, 663, 693, 748, 828, 837, 870, 888, 1008, 1089, 1136, 1192, 1197, 1210, 1288, 1416, 1422, 1440

Offset: 1

S. I. Dimitrov, Jun 07 2025

nonn

Let sigma be the usual sum-of-divisors function. We say that x and y form a cross-harmonious pair if y/sigma(x) + x/sigma(y) = 1. An amicable pair forms a cross-harmonious pair, so the larger member of an amicable pair A002046 is a term of this sequence.

If a < b are Mersenne exponents (A000043) then 2^(a-1) * (2^b - 1) is a term, as it and 2^(b-1) * (2^a - 1) are a cross-harmonious pair. - Robert Israel, Jul 07 2025

			12 and 14 form a cross-harmonious pair since 14/sigma(12) + 12/sigma(14) = 14/28 + 12/24 = 1.

Robert Israel, Table of n, a(n) for n = 1..500
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000043, A000203, A000396, A002025, A002046, A253535, A253534.

N:= 10000: # for terms <= N
S:= map(numtheory:-sigma, [$1..N]):
filter:= proc(x) ormap(y -> y/S[x]+x/S[y]=1, [$1..x-1]) end proc:
select(filter, [$1..N]); # Robert Israel, Jul 07 2025

isok(y) = for (x=1, y-1, if (y/sigma(x) + x/sigma(y) == 1, return(x));); \\ Michel Marcus, Jun 09 2025

cp's OEIS Frontend

A253534 Larger member of a harmonious pair.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A384487 Numbers k such that there exist two integers 0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A384706 Integers y such that there exists an integer 0 < x < y such that y/sigma(x) + x/sigma(y) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI