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

Offset: 1

Author

Griffin N. Macris, Mar 23 2016

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A069262, A139544.

Difference of A214515 and A270781.

Difference of A214515 and A004215.

Programs

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))]])]
[x for x in A if x in T] # Tom Edgar, Mar 24 2016