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A328370 Quasi-amicable 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 quasi-amicable if sigma(x) = sigma(y) = x + y + 1, where sigma(n) is the sum of the divisors of n.
All known quasi-amicable pairs have opposite parity.
First differs from A005276 at a(6).
According to Hisanori Mishima (see link) there are 404 quasi-amicable 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.
P. Hagis and G. Lord, Quasi-amicable numbers, Math. Comp. 31 (1977), 608-611.
Paul Pollack, Quasi-Amicable Numbers are Rare, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.2.
Eric Weisstein's World of Mathematics, Quasiamicable Pair.
FORMULA
a(2*n-1) = A003502(n); a(2*n) = A003503(n).
EXAMPLE
Initial quasi-amicable 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 quasi-amicable 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:
aList(10000); # Peter Luschny, Nov 18 2019
CROSSREFS
A003502 and A003503 interleaved.
Sequence in context: A211721 A057533 A005276 * A143722 A261709 A261704
KEYWORD
nonn
AUTHOR
Omar E. Pol, Oct 14 2019
STATUS
approved

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Last modified March 1 12:27 EST 2024. Contains 370433 sequences. (Running on oeis4.)