cp's OEIS Frontend

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.

%I A270783 #33 Mar 08 2020 06:55:11
%S A270783 16,21,26,36,37,42,52,58,61,66,68,76,82,84,100,106,108,116,132,133,
%T A270783 138,148,154,164,172,178,180,181,186,196,202,204,212,226,228,236,244,
%U A270783 250,260,268,276,292,298,300,301,306,308,322,324,332,340
%N 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.
%C A270783 This sequence is infinite since 4p^2 = 0^2 + 0^2 + (2p)^2 is in the sequence for all primes p.
%C A270783 A069262 is a subsequence.
%C A270783 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 A270783 Griffin N. Macris, <a href="/A270783/b270783.txt">Table of n, a(n) for n = 1..500</a>
%e A270783 a(1) = 16 = 2^2 + 2^2 + 2^2 + 2^2 = 0^2 + 0^2 + 4^2.
%o A270783 (Sage)
%o A270783 n=340 #change for more terms
%o A270783 P=prime_range(1,ceil(sqrt(n)))
%o A270783 S=cartesian_product_iterator([P,P,P,P])
%o A270783 A=list(Set([sum(i^2 for i in y) for y in S if sum(i^2 for i in y)<=n]))
%o A270783 A.sort()
%o A270783 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 A270783 [x for x in A if x in T] # _Tom Edgar_, Mar 24 2016
%Y A270783 Cf. A069262, A139544.
%Y A270783 Difference of A214515 and A270781.
%Y A270783 Difference of A214515 and A004215.
%K A270783 nonn
%O A270783 1,1
%A A270783 _Griffin N. Macris_, Mar 23 2016