%I
%S 6232,6368,10744,10856,12285,14595,66928,66992,67095,71145,79750,
%T 88730,100485,124155,122265,139815,122368,123152,141664,153176,142310,
%U 168730,176272,180848,185368,203432,356408,399592,437456,455344,522405,525915,600392,669688,609928,686072
%N List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.
%C The sequence lists those amicable pairs (m,n) in increasing order where the sum of the amicable pair is divisible by ten.
%C Up to the first 5001 amicable pairs, 88.1% of the sums fulfill this condition (up to the first 100 amicable pairs: 73%; up to the first 1000: 82.5%; up to 2000: 85.5%). So the conjecture here is that as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100%.
%D R. K. Guy, Unsolved Problems in Number Theory, SpringerVerlag, 1994, pp. 55  58.
%D Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 67  69.
%H Zoltan Galantai, <a href="/A291422/a291422_2.txt">List of the amicable pairs where the sum divisible by ten</a> smaller and larger amicable numbers; sums (the first 4406 pairs)
%H Zoltan Galantai, <a href="/A291422/a291422_3.txt">List of the first 5001 amicable pairs with their sums</a> denoting whether the sum is divisible by ten or not
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AmicablePair.html">Amicable Pair</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SociableNumbers.html">Sociable Numbers</a>.
%e The sum of 6232 and 6368 is divisible by ten, thus the (6232, 6368) amicable pair belongs to the sequence. On the other hand, the (220, 284) amicable pair does not qualify since its sum is 504.
%p with(numtheory): P:=proc(q) local a,b,n; for n from 1 to q do a:=sigma(n)n; b:=sigma(a)a;
%p if b=n and a>b and a+b mod 10=0 then print(n); print(a); fi; od; end: P(10^6); # _Paolo P. Lava_, Aug 24 2017
%o (PARI) lista(nn) = {for (n=1, nn, spd = sigma(n)n; if ((spd > n) && (sigma(spd)spd == n) && !((n + spd) % 10), print1(n, ", ", spd, ", ")););} \\ _Michel Marcus_, Aug 26 2017
%Y Cf. A002025, A002046, A063990, A180164, A259180, A259933, A291550.
%K nonn,tabf
%O 1,1
%A _Zoltan Galantai_, Aug 22 2017
