A270783 - OEIS

A270783

Numbers of the form p^2 + q^2 + r^2 + s^2 = a^2 + b^2 + c^2 for some primes p, q, r, and s and some integers a, b, and c.

16, 21, 26, 36, 37, 42, 52, 58, 61, 66, 68, 76, 82, 84, 100, 106, 108, 116, 132, 133, 138, 148, 154, 164, 172, 178, 180, 181, 186, 196, 202, 204, 212, 226, 228, 236, 244, 250, 260, 268, 276, 292, 298, 300, 301, 306, 308, 322, 324, 332, 340

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OFFSET

1,1

COMMENTS

This sequence is infinite since 4p^2 = 0^2 + 0^2 + (2p)^2 is in the sequence for all primes p.

A069262 is a subsequence.

It appears at first that the squares of A139544(n) for n >= 3 are a subsequence. n=22 is the first counterexample, where A139544(22)^2 = 6084 is not an element of this sequence.

LINKS

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

EXAMPLE

a(1) = 16 = 2^2 + 2^2 + 2^2 + 2^2 = 0^2 + 0^2 + 4^2.

PROG

(Sage)

n=340 #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))]])]