A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10.

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

Offset: 1

Author

Zoltan Galantai, Aug 22 2017

Keywords

• nonn
• tabf

Comments

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

Examples

			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.
		

References

• 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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Sergei Chernykh, Amicable pairs list.
• Zoltan Galantai, List of the amicable pairs where the sum divisible by ten smaller and larger amicable numbers; sums (the first 4406 pairs).
• Zoltan Galantai, List of the first 5001 amicable pairs with their sums denoting whether the sum is divisible by ten or not.
• Eric Weisstein's World of Mathematics, Amicable Pair.
• Eric Weisstein's World of Mathematics, Sociable Numbers.

Crossrefs

Cf. A002025, A002046, A063990, A180164, A259180, A259933, A291550.

Programs

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