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A270781
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Numbers n with the property that n is both of the form p^2 + q^2 + r^2 + s^2 for some primes p, q, r, and s, and not of the form a^2 + b^2 + c^2 for any integers a, b, and c.
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2
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31, 47, 63, 71, 79, 87, 92, 103, 111, 124, 127, 143, 151, 156, 159, 175, 183, 188, 191, 199, 207, 220, 223, 231, 247, 252, 255, 271, 295, 303, 311, 316, 319, 327, 343, 348, 351, 367, 383, 391, 399, 412, 415, 423, 439, 444, 463, 471, 476, 487
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OFFSET
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1,1
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COMMENTS
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This sequence can easily be shown to be infinite. Take p, q, r equal and congruent to 1 mod 16, and s = 5. Then, because p = 1+16k, n = 28 + 96k + 768k^2, and n = 4*(7+8*m) for m = 3k+24k^2. Then, following from Legendre's three-square theorem, n cannot be written as a^2 + b^2 + c^2 for any a, b, c in the integers. Then, because there are infinitely many primes of the form p = 1+16k, this sequence is infinite.
It appears at first that all Mersenne numbers (A000225) are included in this sequence. However, this is not the case. The first counterexample is 262143 = 2^18 - 1. The next are 4194303 = 2^22 - 1 and 16777215 = 2^24 - 1.
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LINKS
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EXAMPLE
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31 = 2^2 + 3^2 + 3^2 + 3^2, and, according to Legendre's three-square theorem, 31 cannot be expressed as the sum of three squares, so 31 is a term.
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PROG
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(Sage)
n=487 #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 not(x in T)] # Tom Edgar, Mar 24 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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