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A328370 Quasi-amicable pairs.

%I A328370 #33 Feb 16 2025 08:33:58
%S A328370 48,75,140,195,1050,1925,1575,1648,2024,2295,5775,6128,8892,16587,
%T A328370 9504,20735,62744,75495,186615,206504,196664,219975,199760,309135,
%U A328370 266000,507759,312620,549219,526575,544784,573560,817479,587460,1057595,1000824,1902215,1081184,1331967,1139144,1159095,1140020,1763019
%N A328370 Quasi-amicable pairs.
%C A328370 Also called betrothed pairs, or quasiamicable pairs, or reduced amicable pairs.
%C A328370 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.
%C A328370 All known quasi-amicable pairs have opposite parity.
%C A328370 First differs from A005276 at a(6).
%C A328370 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
%H A328370 R. K. Guy, Unsolved Problems in Number Theory, B5.
%H A328370 P. Hagis and G. Lord, <a href="https://doi.org/10.1090/S0025-5718-1977-0434939-3">Quasi-amicable numbers</a>, Math. Comp. 31 (1977), 608-611.
%H A328370 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/math09/math09t2.htm">Table of quasi-amicable pairs under 10^10</a>.
%H A328370 Paul Pollack, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pollack/pollack3.html">Quasi-Amicable Numbers are Rare</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.2.
%H A328370 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuasiamicablePair.html">Quasiamicable Pair.</a>
%F A328370 a(2*n-1) = A003502(n); a(2*n) = A003503(n).
%e A328370 Initial quasi-amicable pairs:
%e A328370     48,   75;
%e A328370    140,  195;
%e A328370   1050, 1925;
%e A328370   1575, 1648;
%e A328370   2024, 2295;
%e A328370 ...
%e A328370 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.
%p A328370 with(numtheory): aList := proc(searchbound)
%p A328370 local r, n, m, L: L := []:
%p A328370 for m from 1 to searchbound do
%p A328370    n := sigma(m) - m - 1:
%p A328370    if n <= m then next fi;
%p A328370    r := sigma(n) - n - 1:
%p A328370    if r = m then L := [op(L), m, n] fi;
%p A328370 od; L end:
%p A328370 aList(10000); # _Peter Luschny_, Nov 18 2019
%Y A328370 A003502 and A003503 interleaved.
%Y A328370 Cf. A000203, A005276, A126160, A179612, A259180.
%K A328370 nonn
%O A328370 1,1
%A A328370 _Omar E. Pol_, Oct 14 2019