A213721 Smallest prime that is the sum of four nonzero squares in exactly n ways, or -1 if there is no such prime.

2, 7, 31, 67, 97, 103, 157, 199, 277, 223, 307, 383, 439, 367, 547, 463, 673, 613, 577, 751, 643, 733, 787, 727, 853, 983, 997, 967, 1171, 1223, 1063, 1087, 1123, 1613, 1237, 1279, 1471, 1453, 1669, 1423, 1483, 1597, 1627, 1543, 1567, 1747, 2039, 1753, 1867, 1951

Offset

0,1

Comments

a(88) > 10^6 if it is not -1. - Robert Israel, Mar 27 2026

Links

Table of n, a(n) for n=0..49.

Index entries for sequences related to sums of squares

Example

a(2) = 31 because 31 = 2*1 + 4 + 25 = 4 + 3*9.

Maple

N:= 10000: # to consider primes <= N
V:= Vector(N):
for i from 1 while 4*i^2 <= N do
  si:= i^2;
  for j from i while si + 3*j^2 <= N do
    sj:= i^2 + j^2;
    for k from j while sj + 2*k^2 <= N do
      sk:= sj + k^2;
      m0:= k + 1 - (k + sk mod 2);
      for m from m0 by 2 do
        v:= sk + m^2; if v > N then break fi;
        V[v]:= V[v]+1;
od od od od:
R:= 'R':
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  v:= V[p];
  if not assigned(R[v]) then R[v]:= p fi;
od:
inds:= map(op, {indices(R)}):
u:= min({$0 .. (1+max(inds))} minus inds):
[seq(R[i], i=0..u-1)]; # Robert Israel, Mar 27 2026

Mathematica

lst = {}; Do[p = 2; While[True, If[PrimeQ[p] && Length@Select[PowersRepresentations[p, 4, 2], ! MemberQ[#, 0] &] == n, AppendTo[lst, p]; Break[]]; p++], {n, 0, 49}]; lst

Crossrefs

Cf. A025416.

Sequence in context: A191073 A049576 A298169 * A102162 A059846 A343532

Adjacent sequences: A213718 A213719 A213720 * A213722 A213723 A213724

Keyword

nonn

Author

Arkadiusz Wesolowski, Nov 02 2012

Extensions

Name edited by Robert Israel, Mar 27 2026

Status

approved