A094712 -id:A094712 - cp's OEIS frontend

A085317 Primes which are the sum of three nonzero squares.

3, 11, 17, 19, 29, 41, 43, 53, 59, 61, 67, 73, 83, 89, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 233, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 373, 379, 389, 397, 401

Offset: 1

Labos Elemer, Jul 01 2003

nonn

This sequence consists of the primes p (not 5, 13, or 37) such that p == 1, 3 or 5 (mod 8). The density of these primes is 0.75. - T. D. Noe, May 21 2004

Primes of the form a^2 + b^2 + c^2 with 1 <= a <= b <= c. - Zak Seidov, Nov 08 2013

			101 is a term since 101 = 64 + 36 + 1 = 8^2 + 6^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000408.
Cf. A094712 (primes that are not the sum of three positive squares).
Cf. A094713 (number of ways that prime(n) can be represented as a^2+b^2+c^2 with a >= b >= c > 0).

lst={}; lim=32; Do[n=a^2+b^2+c^2; If[nHarvey P. Dale, Jun 18 2022 *)

A094713 Number of ways that prime(n) can be represented as a^2+b^2+c^2 with c >= b >= a > 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 0, 1, 2, 1, 1, 0, 1, 0, 2, 3, 1, 3, 0, 2, 1, 2, 0, 3, 2, 2, 3, 0, 1, 1, 0, 3, 3, 2, 0, 1, 2, 0, 2, 0, 3, 2, 3, 0, 3, 4, 4, 0, 5, 0, 1, 5, 2, 4, 2, 0, 2, 2, 2, 2, 3, 3, 4, 0, 0, 2, 2, 0, 5, 1, 5, 4, 5, 2, 0, 3, 0, 3, 5, 2, 7, 0, 4, 0, 0, 5, 2, 0, 7, 8, 3, 2, 2, 4, 5, 8, 3

Offset: 1

T. D. Noe, May 21 2004

nonn

			a(13) = 2 because prime(13) = 41 = 1+4+36 = 9+16+16.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A085317 (primes that are the sum of three positive squares), A094712 (primes that are not the sum of three positive squares), A094714 (least prime having exactly n representations as the sum of three positive squares).

lim=25; pLst=Table[0, {PrimePi[lim^2]}]; Do[n=a^2+b^2+c^2; If[n?(Min[#]>0&)],{n,110}]  (* _Harvey P. Dale, Feb 17 2011 *)

A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

Original entry on oeis.org

1, 4, 9, 10, 16, 18, 20, 25, 26, 27, 28, 33, 34, 36, 40, 42, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 95, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 120, 121, 122, 123, 124, 125

Offset: 1

Jonas Wallgren, Aug 10 2009, Aug 17 2009

nonn

From Franklin T. Adams-Watters, Aug 29 2009: (Start)
The k_i must all be positive integers.
Note that every integer > 33 is the sum of 5 positive squares, and for n > 5, every integer > n+13 is the sum of n positive squares. (End)
The complement of this sequence includes: A000040, A037074, A143206, 2 * A002145, and 3 * A094712. - Robert Israel, Jan 27 2025

			34 = 2*(4^2 + 1^2), 42 = 3*(3^2 + 2^2 + 1^2), thus 34 and 42 are in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290, A000404, A000408, A000414, A047700, A111178. [From Franklin T. Adams-Watters, Aug 29 2009]

Cf. A000040, A002145, A037074, A094712, A143206.

g:= proc(y,m)
  # can we write y as sum of m positive squares?
   option remember;
   local x;
   if y < m then return false fi;
   if m = 1 then return issqr(y) fi;
   if issqr(y-m+1) then return true fi;
   for x from 1 while x^2 + m-1 < y do
     if procname(y-x^2,m-1) then return true fi
   od;
   false
end proc:
filter:= proc(n)
  ormap(t -> g(n/t, t), numtheory:-divisors(n))
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 26 2025

issumsqs(n,k) = if(n<=0||k<=0,return(k==0&&n==0)); forstep(j=sqrtint(n),max(sqrtint(n\k),1),-1,if(issumsqs(n-j^2,k-1),return(1)));0
isa(n)=local(ds);ds=divisors(n);for(k=1,(#ds+1)\2,if(issumsqs(n\ds[k],ds[k]),return(1)));0
for(n=1,200,if(isa(n),print1(n","))) \\ Franklin T. Adams-Watters, Aug 29 2009

More terms from Franklin T. Adams-Watters, Aug 29 2009

cp's OEIS Frontend

A085317 Primes which are the sum of three nonzero squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A094713 Number of ways that prime(n) can be represented as a^2+b^2+c^2 with c >= b >= a > 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions