I was prompted to investigate this one when I was checking the OEIS for the prime number 24329 that arose from my daily count of the number of days that I've been breathing in Earth's air. One entry reported that 24329 is a member of sequence A100559 that lists the smallest prime equal to the sum of n distinct squares. 24329 appeared in the case of n=41 so this prime was the smallest one that was equal to the sum of 41 distinct squares. The members of the sequence, up to 24329, are:
5, 29, 71, 79, 131, 179, 269, 349, 457, 569, 719, 971, 1171, 1327, 1601, 1913, 2269, 2593, 2999, 3539, 4099, 4549, 5231, 5717, 6529, 7297, 7879, 8779, 9791, 10711, 11867, 12809, 14081, 15269, 16561, 17863, 19463, 20771, 22541, 24329
The sequence begins with 5 for the case of n=2 where \(1^2+2^2=5\). The examples shown in the OEIS entry are as follows:
\(a(3)=29 \text{ because } 29=2^2+3^2+4^2\)
\(a(4) = 71 = 1^2+3^2+5^2+6^2\)
\(a(5)=79 \text{ because } 79=1^2+2^2+3^2+4^2+7^2\)
It can be seen that the squared numbers do not necessarily need to include 1 nor all of the numbers between 1 and n. It can even include numbers greater than n. For example, 29 (when n=3) does not include 1; 71 (when n=4) begins with 1 but does not include 4 but includes 5 and 6; 79 (when n=5) includes 1, 2, 3, 4 and 7 (but not 5 or 6).
Interestingly, if the squares of the first n numbers are added together, they do not add to a prime number for the cases n=2 to 42. It may be a mathematical fact that the sum of the squares of the integers in sequence can never add to a prime number. In the list below, I've worked out these sums in Excel and put them in the central column with the corresponding primes from OEIS A100559 in the right-hand column:
\(a(4) = 71 = 1^2+3^2+5^2+6^2\)
\(a(5)=79 \text{ because } 79=1^2+2^2+3^2+4^2+7^2\)
It can be seen that the squared numbers do not necessarily need to include 1 nor all of the numbers between 1 and n. It can even include numbers greater than n. For example, 29 (when n=3) does not include 1; 71 (when n=4) begins with 1 but does not include 4 but includes 5 and 6; 79 (when n=5) includes 1, 2, 3, 4 and 7 (but not 5 or 6).
Interestingly, if the squares of the first n numbers are added together, they do not add to a prime number for the cases n=2 to 42. It may be a mathematical fact that the sum of the squares of the integers in sequence can never add to a prime number. In the list below, I've worked out these sums in Excel and put them in the central column with the corresponding primes from OEIS A100559 in the right-hand column:
1 | 1 | |
2 | 5 | 5 |
3 | 14 | 29 |
4 | 30 | 71 |
5 | 55 | 79 |
6 | 91 | 131 |
7 | 140 | 179 |
8 | 204 | 269 |
9 | 285 | 349 |
10 | 385 | 457 |
11 | 506 | 569 |
12 | 650 | 719 |
13 | 819 | 971 |
14 | 1015 | 1171 |
15 | 1240 | 1327 |
16 | 1496 | 1601 |
17 | 1785 | 1913 |
18 | 2109 | 2269 |
19 | 2470 | 2593 |
20 | 2870 | 2999 |
21 | 3311 | 3539 |
22 | 3795 | 4099 |
23 | 4324 | 4549 |
24 | 4900 | 5231 |
25 | 5525 | 5717 |
26 | 6201 | 6529 |
27 | 6930 | 7297 |
28 | 7714 | 7879 |
29 | 8555 | 8779 |
30 | 9455 | 9791 |
31 | 10416 | 10711 |
32 | 11440 | 11867 |
33 | 12529 | 12809 |
34 | 13685 | 14081 |
35 | 14910 | 15269 |
36 | 16206 | 16561 |
37 | 17575 | 17863 |
38 | 19019 | 19463 |
39 | 20540 | 20771 |
40 | 22140 | 22541 |
41 | 23821 | 24329 |
42 | 25585 | 25913 |
It can be seen that the primes are always larger than the consecutive sums of squares to the left but not generally larger than the next sum (except in the case n=4 where 71 is larger than 55, the sum of the first 5 squares).
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