A270781 - OEIS

A270781

Numbers n with the property that n is both of the form p^2 + q^2 + r^2 + s^2 for some primes p, q, r, and s, and not of the form a^2 + b^2 + c^2 for any integers a, b, and c.

31, 47, 63, 71, 79, 87, 92, 103, 111, 124, 127, 143, 151, 156, 159, 175, 183, 188, 191, 199, 207, 220, 223, 231, 247, 252, 255, 271, 295, 303, 311, 316, 319, 327, 343, 348, 351, 367, 383, 391, 399, 412, 415, 423, 439, 444, 463, 471, 476, 487

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OFFSET

1,1

COMMENTS

This sequence can easily be shown to be infinite. Take p, q, r equal and congruent to 1 mod 16, and s = 5. Then, because p = 1+16k, n = 28 + 96k + 768k^2, and n = 4*(7+8*m) for m = 3k+24k^2. Then, following from Legendre's three-square theorem, n cannot be written as a^2 + b^2 + c^2 for any a, b, c in the integers. Then, because there are infinitely many primes of the form p = 1+16k, this sequence is infinite.

It appears at first that all Mersenne numbers (A000225) are included in this sequence. However, this is not the case. The first counterexample is 262143 = 2^18 - 1. The next are 4194303 = 2^22 - 1 and 16777215 = 2^24 - 1.

LINKS

Griffin N. Macris, Table of n, a(n) for n = 1..500

EXAMPLE

31 = 2^2 + 3^2 + 3^2 + 3^2, and, according to Legendre's three-square theorem, 31 cannot be expressed as the sum of three squares, so 31 is a term.

PROG

(Sage)

n=487 #change for more terms

P=prime_range(1, ceil(sqrt(n)))

S=cartesian_product_iterator([P, P, P, P])

A=list(Set([sum(i^2 for i in y) for y in S if sum(i^2 for i in y)<=n]))

A.sort()

T=[sum(i^2 for i in y) for y in cartesian_product_iterator([[0..ceil(sqrt(n))], [0..ceil(sqrt(n))], [0..ceil(sqrt(n))]])]