%I #33 Mar 08 2020 06:55:11

%S 16,21,26,36,37,42,52,58,61,66,68,76,82,84,100,106,108,116,132,133,

%T 138,148,154,164,172,178,180,181,186,196,202,204,212,226,228,236,244,

%U 250,260,268,276,292,298,300,301,306,308,322,324,332,340

%N 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.

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

%C A069262 is a subsequence.

%C 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.

%H Griffin N. Macris, <a href="/A270783/b270783.txt">Table of n, a(n) for n = 1..500</a>

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

%o (Sage)

%o n=340 #change for more terms

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

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

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

%o A.sort()

%o 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))]])]

%o [x for x in A if x in T] # _Tom Edgar_, Mar 24 2016

%Y Cf. A069262, A139544.

%Y Difference of A214515 and A270781.

%Y Difference of A214515 and A004215.

%K nonn

%O 1,1

%A _Griffin N. Macris_, Mar 23 2016