

A328370


Quasiamicable pairs.


0



48, 75, 140, 195, 1050, 1925, 1575, 1648, 2024, 2295, 5775, 6128, 8892, 16587, 9504, 20735, 62744, 75495, 186615, 206504, 196664, 219975, 199760, 309135, 266000, 507759, 312620, 549219, 526575, 544784, 573560, 817479, 587460, 1057595, 1000824, 1902215, 1081184, 1331967, 1139144, 1159095, 1140020, 1763019
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Also called betrothed pairs, or quasiamicable pairs, or reduced amicable pairs.
A pair of numbers x and y is called quasiamicable if sigma(x) = sigma(y) = x + y + 1, where sigma(n) is the sum of the divisors of n.
All known quasiamicable pairs have opposite parity.
First differs from A005276 at a(6).
According to Hisanori Mishima (see link) there are 404 quasiamicable pairs where the smaller part is less than 10^10. See A126160 for more values.  Peter Luschny, Nov 18 2019


LINKS

R. K. Guy, Unsolved Problems in Number Theory, B5.


FORMULA



EXAMPLE

Initial quasiamicable pairs:
48, 75;
140, 195;
1050, 1925;
1575, 1648;
2024, 2295;
...
The sum of the divisors of 48 is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 + 48 = 124. On the other hand the sum of the divisors of 75 is 1 + 3 + 5 + 15 + 25 + 75 = 124. Note that 48 + 75 + 1 = sigma(48) = sigma(75) = 124. The smallest quasiamicable pair is (48, 75), so a(1) = 48 and a(2) = 75.


MAPLE

with(numtheory): aList := proc(searchbound)
local r, n, m, L: L := []:
for m from 1 to searchbound do
n := sigma(m)  m  1:
if n <= m then next fi;
r := sigma(n)  n  1:
if r = m then L := [op(L), m, n] fi;
od; L end:


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



