Number Patterns

My aim is to find sets of integers, A and B, such that

(1) A = (a1, a2, ..., ai) and B = (b1, b2, ..., bj)
(a1) + (a2) + ... + (ai) = (b1) + (b2) + ... + (bj)
(a1)^2 + (a2)^2 + ... + (ai)^2 = (b1)^2 + (b2)^2 + ... + (bj)^2
(a1)^3 + (a2)^3 + ... + (ai)^3 = (b1)^3 + (b2)^3 + ... + (bj)^3
(a1)^4 + (a2)^4 + ... + (ai)^4 = (b1)^4 + (b2)^4 + ... + (bj)^4
(a1)^5 + (a2)^5 + ... + (ai)^5 = (b1)^5 + (b2)^5 + ... + (bj)^5
and,
(a1 + 1)^2 + (a2 + 1)^2 + ... + (ai + 1)^2 = (b1 + 1)^2 + (b2 + 1)^2 + ... + (bj + 1)^2
(a1 + 2)^2 + (a2 + 2)^2 + ... + (ai + 2)^2 = (b1 + 2)^2 + (b2 + 2)^2 + ... + (bj + 2)^2
(a1 + 3)^2 + (a2 + 3)^2 + ... + (ai + 3)^2 = (b1 + 3)^2 + (b2 + 3)^2 + ... + (bj + 3)^2
(a1 + 4)^2 + (a2 + 4)^2 + ... + (ai + 4)^2 = (b1 + 4)^2 + (b2 + 4)^2 + ... + (bj + 4)^2

(2) A = (a1, a2, ..., ai) and B = (b1, b2, ..., bj)
(a1 + p)^2 + (a2 + p)^2 + ... + (ai + p)^2 = (b1 + p)^2 + (b2 + p)^2 + ... + (bj + p)^2
p being the first prime numbers 2, 3, 5, 7


Here are set of equations in which the sums of powers of two different sets of numbers are the same for several different exponents. The simplest example :

1 + 6 + 8 = 15 = 2 + 4 + 9

1^2 + 6^2 + 8^2 = 101 = 2^2 + 4^2 + 9^2

The two sets, here, are: (1, 6, 8) and (2, 4, 9)

1 + 8 + 10 + 17 = 36 = 2 + 5 + 13 + 16

1^2 + 8^2 + 10^2 + 17^2 = 454 = 2^2 + 5^2 + 13^2 + 16^2

1^3 + 8^3 + 10^3 + 17^3 = 6426 = 2^3 + 5^3 + 13^3 + 16^3

The two sets, here, are: (1, 8, 10, 17) and (2, 5, 13, 16)

Remarkably, if any integer is added to all terms of the equation, it will still hold
Adding 1 to the example above, gives

2 + 9 + 11 + 18 = 40 = 3 + 6 + 14 + 17

2^2 + 9^2 + 11^2 + 18^2 = 530 = 3^2 + 6^2 + 14^2 + 17^2

2^3 + 9^3 + 11^3 + 18^3 = 7900 = 3^3 + 6^3 + 14^3 + 17^3

Adding 2 and 3 to all terms of the equation 1 + 8 + 10 + 17 = 36 = 2 + 5 + 13 + 16

we get,

(1+1)^2 + (8+1)^2 + (10+1)^2 + (17+1)^2 = 530

(2+1)^2 + (5+1)^2 + (13+1)^2 + (16+1)^2 = 530

(1+2)^2 + (8+2)^2 + (10+2)^2 + (17+2)^2 = 614
(2+2)^2 + (5+2)^2 + (13+2)^2 + (16+2)^2 = 614

(1+3)^2 + (8+3)^2 + (10+3)^2 + (17+3)^2 = 706
(2+3)^2 + (5+3)^2 + (13+3)^2 + (16+3)^2 = 706

(1+4)^2 + (8+4)^2 + (10+4)^2 + (17+4)^2 = 806
(2+4)^2 + (5+4)^2 + (13+4)^2 + (16+4)^2 = 806

(1+5)^2 + (8+5)^2 + (10+5)^2 + (17+5)^2 = 914

(2+5)^2 + (5+5)^2 + (13+5)^2 + (16+5)^2 = 914

OR

1^2 + 8^2 + 10^2 + 17^2 = 454 = 2^2 + 5^2 + 13^2 + 16^2

2^2 + 9^2 + 11^2 + 18^2 = 530 = 3^2 + 6^2 + 14^2 + 17^2

3^2 + 10^2 + 12^2 + 19^2 = 614 = 4^2 + 7^2 + 15^2 + 18^2

4^2 + 11^2 + 13^2 + 20^2 = 706 = 5^2 + 8^2 + 16^2 + 19^2
5^2 + 12^2 + 14^2 + 21^2 = 806 = 6^2 + 9^2 + 17^2 + 20^2

6^2 + 13^2 + 15^2 + 22^2 = 914 = 7^2 + 10^2 + 18^2 + 21^2

AND

1 + 50 + 57 + 15 + 22 + 71 = 216 = 2 + 45 + 61 + 11 + 27 + 70
1^2 + 50^2 + 57^2 + 15^2 + 22^2 + 71^2 = 11500
2^2 + 45^2 + 61^2 + 11^2 + 27^2 + 70^2 = 11500
1^3 + 50^3 + 57^3 + 15^3 + 22^3 + 71^3 = 682128
2^3 + 45^3 + 61^3 + 11^3 + 27^3 + 70^3 = 682128

1 + 50 + 57 + 15 + 22 + 71 = 2 + 45 + 61 + 11 + 27 + 70

1^2 + 50^2 + 57^2 + 15^2 + 22^2 + 71^2 = 2^2 + 45^2 + 61^2 + 11^2 + 27^2 + 70^2
1^3 + 50^3 + 57^3 + 15^3 + 22^3 + 71^3 = 2^3 + 45^3 + 61^3 + 11^3 + 27^3 + 70^3

AND

1 + 50 + 57 + 15 + 22 + 71 = 216 = 2 + 45 + 61 + 11 + 27 + 70

2^2 + 51^2 + 58^2 + 16^2 + 23^2 + 72^2 = 11938
3^2 + 46^2 + 62^2 + 12^2 + 28^2 + 71^2 = 11938

3^2 + 52^2 + 59^2 + 17^2 + 24^2 + 73^2 = 12388
4^2 + 47^2 + 63^2 + 13^2 + 29^2 + 72^2 = 12388

4^2 + 53^2 + 60^2 + 18^2 + 25^2 + 74^2 = 12850
5^2 + 48^2 + 64^2 + 14^2 + 30^2 + 73^2 = 12850

So,

1 + 50 + 57 + 15 + 22 + 71 = 2 + 45 + 61 + 11 + 27 + 70

1^2 + 50^2 + 57^2 + 15^2 + 22^2 + 71^2 = 2^2 + 45^2 + 61^2 + 11^2 + 27^2 + 70^2

2^2 + 51^2 + 58^2 + 16^2 + 23^2 + 72^2 = 3^2 + 46^2 + 62^2 + 12^2 + 28^2 + 71^2

3^2 + 52^2 + 59^2 + 17^2 + 24^2 + 73^2 = 4^2 + 47^2 + 63^2 + 13^2 + 29^2 + 72^2

4^2 + 53^2 + 60^2 + 18^2 + 25^2 + 74^2 = 5^2 + 48^2 + 64^2 + 14^2 + 30^2 + 73^2

5^2 + 54^2 + 61^2 + 19^2 + 26^2 + 75^2 = 6^2 + 49^2 + 65^2 + 15^2 + 31^2 + 74^2

6^2 + 55^2 + 62^2 + 20^2 + 27^2 + 76^2 = 7^2 + 50^2 + 66^2 + 16^2 + 32^2 + 75^2

5 + 37 + 66 + 6 + 35 + 67 = 216
1 + 50 + 57 + 15 + 22 + 71 = 2 + 45 + 61 + 11 + 27 + 70 = 216
So 1 + 50 + 57 + 15 + 22 + 71 = 2 + 45 + 61 + 11 + 27 + 70 = 5 + 37 + 66 + 6 + 35 + 67

5^2 + 37^2 + 66^2 + 6^2 + 35^2 + 67^2 = 11500
5^3 + 37^3 + 66^3 + 6^3 + 35^3 + 67^3 = 682128

So,

1^2 + 50^2 + 57^2 + 15^2 + 22^2 + 71^2
= 2^2 + 45^2 + 61^2 + 11^2 + 27^2 + 70^2
= 5^2 + 37^2 + 66^2 + 6^2 + 35^2 + 67^2

1^3 + 50^3 + 57^3 + 15^3 + 22^3 + 71^3
= 2^3 + 45^3 + 61^3 + 11^3 + 27^3 + 70^3
= 5^3 + 37^3 + 66^3 + 6^3 + 35^3 + 67^3

We have 3 sets of integers
(1, 50, 57, 15, 22, 71), (2, 45, 61, 11, 27, 70), (5, 37, 66, 6, 35, 67)

[To Be Continued]