A103381 Numbers which are the sum of four distinct squares, a^2+b^2+c^2+d^2, such that a^2+b^2, c^2+d^2, a^2+d^2 and b^2+c^2 are all prime.

46, 66, 78, 102, 114, 126, 142, 154, 162, 174, 198, 222, 246, 262, 270, 274, 282, 286, 294, 298, 318, 322, 334, 342, 354, 366, 378, 382, 394, 402, 406, 414, 426, 438, 442, 454, 462, 486, 498, 502, 510, 518, 522, 526, 534, 538, 546, 550, 558, 562, 574, 582

Offset

1,1

Comments

Such primes must be == 1 mod 4. All the terms of the sequence are of the form 4*k+2.

Links

Table of n, a(n) for n=1..52.

Example

46 = 1^2+2^2+5^2+4^2 = 5+41 = 17+29
102 = 3^2+2^2+5^2+8^2 = 13+89 = 73+29

Mathematica

t = Partition[ Flatten[ Table[{a^2 + b^2 + c^2 + d^2, a^2, b^2, c^2, d^2}, {d, 4, 20}, {c, 3, d - 1}, {b, 2, c - 1}, {a, b - 1}]], 5]; lst = {}; lg = Length[t]; Do[a = t[[n, 2]]; b = t[[n, 3]]; c = t[[n, 4]]; d = t[[n, 5]]; If[ Drop[ Sort[ Join[ {Mod[a + b, 4] == 1 && Mod[c + d, 4] == 1 && PrimeQ[a + b] && PrimeQ[c + d]}, {Mod[a + c, 4] == 1 && Mod[b + d, 4] == 1 && PrimeQ[a + c] && PrimeQ[b + d]}, {Mod[a + d, 4] == 1 && Mod[b + c, 4] == 1 && PrimeQ[a + d] && PrimeQ[b + c]}]], 1] == {True, True}, AppendTo[ lst, t[[n, 1]] ]], {n, lg}]; Take[ Union[ lst], 52] (* Robert G. Wilson v, Mar 25 2005 *)
apQ[{a_, b_, c_, d_}]:=AllTrue[{a+b, c+d, a+d, b+c}, PrimeQ]; Total/@Select[ Flatten[ Permutations/@Subsets[Range[20]^2, {4}], 1], apQ]//Union (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 22 2018 *)

Crossrefs

Sequence in context: A020175 A118698 A350203 * A101577 A051413 A167283

Adjacent sequences: A103378 A103379 A103380 * A103382 A103383 A103384

Keyword

nonn

Author

Robin Garcia, Mar 20 2005

Extensions

Edited by Don Reble and Robert G. Wilson v, Mar 25 2005

Status

approved