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A291422 List of pairs of amicable numbers (m,n) where the sum of the pair is divisible by 10. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 fulfill 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

REFERENCES

R. 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

Table of n, a(n) for n=1..36.

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.

EXAMPLE

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.

MAPLE

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

PROG

(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

CROSSREFS

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

Sequence in context: A112665 A232390 A092726 * A200830 A028545 A173943

Adjacent sequences:  A291419 A291420 A291421 * A291423 A291424 A291425

KEYWORD

nonn,tabf

AUTHOR

Zoltan Galantai, Aug 22 2017

STATUS

approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)