

A276533


Least prime p with A271518(p) = n.


2



5, 2, 19, 127, 17, 67, 163, 41, 89, 101, 131, 313, 257, 211, 227, 461, 241, 401, 613, 337, 433, 353, 577, 467, 863, 887, 617, 787, 601, 569, 761, 641, 823, 673, 857, 1217, 881, 1091, 1289, 977, 1427, 1097, 1801, 929, 1153, 953, 1321, 1049, 1747, 1409
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OFFSET

1,1


COMMENTS

Conjecture: a(n) exists for any positive integer n.
In contrast, it is known that for each prime p the number of ordered integral solutions to the equation x^2 + y^2 + z^2 + w^2 = p is 8*(p+1).
In 1998 J. Friedlander and H. Iwaniec proved that there are infinitely many primes p of the form w^2 + x^4 = w^2 + (x^2)^2 + 0^2 + 0^2 with w and x nonnegative integers. Since x^2 + 3*0 + 5*0 is a square, we see that A271518(p) > 0 for infinitely many primes p.


REFERENCES

J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, Ann. of Math. 148 (1998), 9451040.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..500
ZhiWei Sun, Refining Lagrange's foursquare theorem, arXiv:1604.06723 [math.GM], 2016.


EXAMPLE

a(1) = 5 since 5 is the first prime which can be written in a unique way as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integer and x + 3*y + 5*z a square; in fact, 5 = 1^2 + 0^2 + 0^2 + 2^2 with 1 + 3*0 + 5*0 = 1^2.
a(2) = 2 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 3*0 + 5*0 = 1^2, and 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 + 3*1 + 5*0 = 2^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[m=0; Label[aa]; m=m+1; r=0; Do[If[SQ[Prime[m]x^2y^2z^2]&&SQ[x+3y+5z], r=r+1; If[r>n, Goto[aa]]], {x, 0, Sqrt[Prime[m]]}, {y, 0, Sqrt[Prime[m]x^2]}, {z, 0, Sqrt[Prime[m]x^2y^2]}]; If[r<n, Goto[aa], Print[n, " ", Prime[m]]]; Continue, {n, 1, 50}]


CROSSREFS

Cf. A000040, A000118, A000290, A028916, A271518, A273294, A273302, A278560.
Sequence in context: A286154 A304635 A306198 * A303685 A189746 A191667
Adjacent sequences: A276530 A276531 A276532 * A276534 A276535 A276536


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 12 2016


STATUS

approved



