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

1,1

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

Crossrefs

Cf. A069262, A139544.

Difference of A214515 and A270781.

Difference of A214515 and A004215.

Sequence in context: A050436 A038868 A214515 * A330591 A302167 A186454

Adjacent sequences: A270780 A270781 A270782 * A270784 A270785 A270786

Keyword

nonn

Author

Griffin N. Macris, Mar 23 2016

Status

approved