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A291422
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List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.
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4
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6232, 6368, 10744, 10856, 12285, 14595, 66928, 66992, 67095, 71145, 79750, 88730, 100485, 124155, 122265, 139815, 122368, 123152, 141664, 153176, 142310, 168730, 176272, 180848, 185368, 203432, 356408, 399592, 437456, 455344, 522405, 525915, 600392, 669688, 609928, 686072
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refs;
listen;
history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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The sequence lists those amicable pairs (m,n) in increasing order where the sum of the amicable pair is divisible by ten.
Up to the first 5001 amicable pairs, 88.1% of the sums satisfy this condition (up to the first 100 amicable pairs: 74%; up to the first 1000: 82.5%; up to 2000: 85.25%). 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%. [corrected by Paul Zimmermann, Feb 05-06 2019]
Among the 1947667 pairs up to 19 digits from Sergei Chernykh's database, there are 1872573 pairs with m+n divisible by ten, thus about 96.14%. - Paul Zimmermann, Feb 07 2019
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 55-58.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 67-69.
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LINKS
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EXAMPLE
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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.
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MAPLE
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with(numtheory): P:=proc(q) local a, b, n; for n from 1 to q do a:=sigma(n)-n; b:=sigma(a)-a;
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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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