

A291624


Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 2*y + 5*z, p  2 and p + 4 are all prime.


2



0, 1, 1, 0, 1, 3, 1, 0, 1, 2, 2, 0, 3, 7, 3, 0, 4, 4, 1, 0, 4, 7, 3, 0, 3, 5, 2, 0, 4, 6, 2, 0, 2, 3, 3, 0, 4, 8, 3, 0, 5, 8, 2, 0, 2, 5, 2, 0, 5, 8, 4, 0, 4, 5, 2, 0, 5, 6, 4, 0, 1, 8, 5, 0, 3, 9, 3, 0, 6, 8, 3, 0, 5, 13, 5, 0, 9, 9, 2, 0, 4, 6, 6, 0, 7, 11, 4, 0, 8, 10, 5, 0, 2, 11, 5, 0, 3, 10, 4, 0
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OFFSET

1,6


COMMENTS

Conjecture: a(n) > 0 for all n > 1 not divisible by 4.
See also A291635 for a stronger conjecture.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.


EXAMPLE

a(2) = 1 since 2 = 0^2 + 1^2 + 1^2 + 0^2 with 0 + 2*1 + 5*1 = 7, 7  2 = 5 and 7 + 4 = 11 all prime.
a(5) = 1 since 5 = 2^2 + 0^2 + 1^2 + 0^2 with 2 + 2*0 + 5*1 = 7, 7  2 = 5 and 7 + 4 = 11 all prime.
a(181) = 1 since 181 = 1^2 + 6^2 + 0^2 + 12^2 with 1 + 2*6 + 5*0 = 13, 13  2 = 11 and 13 + 4 = 17 all prime.
a(285) = 1 since 285 = 10^2 + 4^2 + 5^2 + 12^2 with 10 + 2*4 + 5*5 = 43, 43  2 = 41 and 43 + 4 = 47 all prime.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
TQ[p_]:=TQ[p]=PrimeQ[p]&&PrimeQ[p2]&&PrimeQ[p+4];
Do[r=0; Do[If[SQ[nx^2y^2z^2]&&TQ[x+2y+5z], r=r+1], {x, 0, Sqrt[n]}, {y, 0, Sqrt[nx^2]}, {z, 0, Sqrt[nx^2y^2]}]; Print[n, " ", r], {n, 1, 100}]


CROSSREFS

Cf. A000040, A000118, A000290, A022004, A271518, A281976, A290935, A291150, A291191, A291455, A291635.
Sequence in context: A328747 A053382 A031253 * A291635 A308243 A268386
Adjacent sequences: A291621 A291622 A291623 * A291625 A291626 A291627


KEYWORD

nonn,look


AUTHOR

ZhiWei Sun, Aug 28 2017


STATUS

approved



