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A253472 Square Pairs: Numbers n such that 1, 2, ..., 2n can be partitioned into n pairs, where each pair adds up to a perfect square. 3
4, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Kiran Kedlaya proved that all numbers greater than 11 are included in the sequence. Outline of proof:

- Show by hand or by computer that it works up to n = 30.

- For n=31, pair 62+59=61+60=11^2 and then reduce to the case of n=29. For n=32, pair 64+57, ..., 61+60 and reduce to the case of 28. And so on. This works until n=48.

- For n=49, ..., 72 pairs adding up to 13^2 allow us to reduce to n=35.

- Repeat the process, always terminating at (2m+1)^2-25, aiming for sums of (2m+3)^2. The first such pair is (2m+1)^2-23, 8m+31.

- This always works, as long as (2m+1)^2 - 25 > 8m+31, and therefore we must have m > 4.

A similar sequence using odd numbers can be created, by making n pairs that sum to perfect squares, using numbers from 0 to 2n-1. All numbers greater than 6 are included.

Worthy of consideration for the elementary school classroom working on square numbers. - Gordon Hamilton, Mar 20 2015

REFERENCES

Alfred S. Posamentier, Stephen Krulik, Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12, 2008, page 191.

LINKS

Table of n, a(n) for n=1..73.

Gordon Hamilton, Kiran S. Kedlaya, and Henri Picciotto, Square-Sum Pair Partitions, College Mathematics Journal 46.4 (2015): 264-269.

Thomas Kilkelly, The ARML Power Contest, 2015, page 77.

Henri Picciotto, Python program to generate sequence

EXAMPLE

For n = 4: (8, 1), (7, 2), (6, 3), (5, 4).

For n = 7: (14, 2), (13, 3), (12, 4), (11, 5), (10, 6), (9, 7), (8, 1).

PROG

(Python) # See link.

CROSSREFS

Cf. A252897.

Sequence in context: A262874 A161986 A020670 * A255060 A047538 A074231

Adjacent sequences:  A253469 A253470 A253471 * A253473 A253474 A253475

KEYWORD

nonn

AUTHOR

Henri Picciotto, Jan 01 2015

STATUS

approved

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Last modified May 24 21:22 EDT 2017. Contains 287008 sequences.