A000414 -id:A000414 - cp's OEIS frontend

A003386 Numbers that are the sum of 8 nonzero 8th powers.

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

Incorrect program removed by David A. Corneth, Aug 01 2020

A097203 Number of 4-tuples (a,b,c,d) with 1 <= a <= b <= c <= d, a^2+b^2+c^2+d^2 = n and gcd(a,b,c,d) = 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 2, 1, 2, 0, 0, 3, 2, 1, 2, 1, 2, 0, 2, 2, 1, 3, 1, 2, 3, 0, 2, 4, 1, 2, 2, 1, 3, 0, 1, 3, 3, 2, 2, 4, 2, 0, 3, 2, 3, 3, 2, 3, 3, 0, 2, 5, 2, 3, 3, 2, 4, 0, 1, 5, 4, 2, 4, 2, 3, 0, 4, 4, 3

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Oct 22 2008

nonn

The old entry with this sequence number was a duplicate of A034836.
From Wolfdieter Lang, Mar 25 2013: (Start)
a(n) = 0 if n has no partition with four parts, each a (nonzero) square, and the parts have no common factor > 1.
n is not representable as a primitive sum of four nonzero squares.
If n' has a representation [s(1),s(2),s(3),s(4)] with 1 <= s(1) <= s(2) <= s(3) <= s(4) and sum(s(j)^2,j=1..4) = n', then [k*s(1),k*s(2),k*s(3),k*s(4)] is a representation of n := k^2*n'. Therefore, only primitive representations with gcd(s(1),s(2),s(3),s(4)) = 1 are here considered.
See A025428(n) for the multiplicity of the representations of n as a sum of four nonzero squares.
For the n values with a(n) not zero (primitively representable as a sum of four nonzero squares) see A222949. (End)

			The solutions (if any) for n <= 20 are as follows:
n = 1:
n = 2:
n = 3:
n = 4: 1 1 1 1
n = 5:
n = 6:
n = 7: 1 1 1 2
n = 8:
n = 9:
n = 10: 1 1 2 2
n = 11:
n = 12: 1 1 1 3
n = 13: 1 2 2 2
n = 14:
n = 15: 1 1 2 3
n = 16:
n = 17:
n = 18: 1 2 2 3
n = 19: 1 1 1 4
n = 20: 1 1 3 3
From _Wolfdieter Lang_, Mar 25 2013: (Start)
a(16) = 0 because 16 is not a primitive sum of four nonzero squares. The representation [2,2,2,2] of 16 is not primitive.
a(40) = 0 because the only representation as sum of four nonzero squares (A025428(4) = 1) is [2,2,4,4], but this is not primitive.
a(28) = 2 because the two primitive representations of 28 are
[1, 1, 1, 5] and [1, 3, 3, 3]. [2, 2, 2, 4] = 2*[1, 1, 1, 2] is not primitive due to 28 = 2^2*7. (End)

N. J. A. Sloane, Vinay Vaishampayan and Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 1..1024 from N. J. A. Sloane and Vinay Vaishampayan)

Cf. A025428, A000414, A222949.

b:= proc(n, i, g, t) option remember; `if`(n=0,
      `if`(g=1 and t=0, 1, 0), `if`(i<1 or t=0 or i^2*tn, 0, b(n-i^2, i, igcd(g, i), t-1))))
    end:
a:= n-> `if`(n<4, 0, b(n, isqrt(n-3), 0, 4)):
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 02 2013

Clear[b]; b[n_, i_, g_, t_] := b[n, i, g, t] = If[n == 0, If[g == 1 && t == 0, 1, 0], If[i < 1 || t == 0 || i^2*t < n, 0, b[n, i-1, g, t] + If[i^2 > n, 0, b[n-i^2, i, GCD[g, i], t-1]]]]; a[n_] := If[n < 4, 0, b[n, Sqrt[n-3] // Floor, 0, 4]]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Apr 05 2013, translated from Alois P. Heinz's Maple program *)

If a(n) > 0 then 8 does not divide n.

a(n) = k if there are k different quadruples [s(1),s(2),s(3),s(4)] with 1 <= s(1) <= s(2) <= s(3) <= s(4), gcd(s(1),s(2),s(3),s(4)) = 1 and sum(s(j)^2,j=1..4) = n. If there is no such quadruple then a(n) = 0. - Wolfdieter Lang, Mar 25 2013

A145202 Primes of form 4n^2 + 4n + 653.

Original entry on oeis.org

653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877, 2333, 2677, 2861, 3253, 3461, 3677, 4133, 4373, 4621, 4877, 5413, 5693, 5981, 6277, 6581, 7213, 7541, 7877, 8221, 8573, 8933, 9677, 10061, 10453, 10853

Offset: 1

Klaus Brockhaus, Oct 04 2008

nonn

First 18 terms are for n from 0 through 17, next terms are for n = 20, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, ...
The sequence of n such that 4*n^2 + 4*n + 653 is composite starts 18, 19, 21, 24, 28, 33, 39, 46, 54, 60, 61, 62, 63, 65, 67, 72, 73, 75, 81, 82, 84, 85, 86, 93, 95, 96, 100, ...
These primes are in A000414. [Bruno Berselli, Apr 20 2014]

			a(18) = 4*17^2 + 4*17 + 653 = 1877.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

A145125 is essentially the same sequence.

Cf. A005846 (primes of form n^2 + n + 41).

[a: n in [0..100] | IsPrime(a) where a is  4*n^2 + 4*n + 653]; // Vincenzo Librandi, Apr 21 2014

Select[Table[4 n^2 + 4 n + 653, {n, 0, 100}], PrimeQ] (* Vincenzo Librandi, Apr 21 2014 *)

{for(n=0, 50, if(isprime(p=4*n^2+4*n+653), print1(p, ",")))}

A347803 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 16, 0, 0, 96, 0, 36, 256, 0, 432, 256, 0, 1728, 64, 486, 2304, 768, 3888, 0, 3072, 7776, 1728, 7112, 0, 13824, 12864, 0, 27648, 6336, 15552, 9261, 18688, 62208, 21744, 24576, 0, 72576, 51456, 24300, 117504, 38400, 101088, 9216, 93184, 155520, 86400, 142382, 62208, 352512, 67344, 0, 202752, 286176

Offset: 0

Seiichi Manyama, Sep 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000414, A014110, A037216, A347801, A347802.

a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(m=1, n, (i^2+j^2+k^2+m^2==n)*(i*j*k*m)^2))));

my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=0, sqrtint(N), k^2*x^k^2)^4))

a(n) is sum of i^2 * j^2 * k^2 *m^2 for positive integers i,j,k,m such that i^2+j^2+k^2+m^2=n.

A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

Original entry on oeis.org

1, 3, 3, 1, 2, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 1, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3

Offset: 1

Yifan Xie, Apr 05 2023

a(n) is the smallest number k such that n*k can be expressed as the sum of k nonzero squares.

			For n = 2, if k = 1, 2*1 = 2 is a nonsquare; if k = 2, 2*2 = 4 cannot be expressed as the sum of 2 nonzero squares; if k = 3, 2*3 = 6 = 2^2+1^2+1^2, so a(2) = 3.

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, p. 140.

Yifan Xie, Table of n, a(n) for n = 1..10000
Wikipedia, Lagrange's four-square theorem
Index entries for sequences related to sums of squares

Cf. A000290, A000378, A000404, A000408, A000414, A000534, A047700.

Cf. A362068 (allows zeros), A362110 (distinct).

findsquare(k, m) = if(k == 1, issquare(m), for(j=1, m, if(j*j+k > m, return(0), if(findsquare(k-1, m-j*j), return(1)))));
a(n) = for(t = 1, n+1, if(findsquare(t, n*t), return(t)));

a(n) <= 4. Proof: With Lagrange's four-square theorem, if 4*n is not the sum of 4 positive squares (see A000534), then it is easy to express 3*n as the sum of 3 positive squares. - Yifan Xie and Thomas Scheuerle, Apr 29 2023

A145125 Primes of the form 4n^2-4n+653.

Original entry on oeis.org

653, 653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877, 2333, 2677, 2861, 3253, 3461, 3677, 4133, 4373, 4621, 4877, 5413, 5693, 5981, 6277, 6581, 7213, 7541, 7877, 8221, 8573, 8933, 9677, 10061, 10453

Offset: 1

Bobby Kramer (panthar1(AT)gmail.com), Oct 02 2008

nonn

These primes are in A000414. [Bruno Berselli, Apr 20 2014]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A145202 is essentially the same sequence.

[ a: n in [0..900] | IsPrime(a) where a is 4*n^2-4*n+653]; // Vincenzo Librandi, Nov 25 2010

Select[Table[4 n^2 - 4 n + 653, {n, 0, 300}], PrimeQ] (* Vincenzo Librandi, Apr 21 2014 *)

More terms from Vincenzo Librandi, Apr 28 2010

A385531 Numbers x such that there exist three integers 00 such that sigma(x)^2 = sigma(y)^2 = sigma(z)^2 = x^2 + y^2 + z^2 + t^2.

Original entry on oeis.org

4, 6, 28, 45, 48, 60, 156, 204, 208, 360, 496, 1170, 2016, 2520, 2925, 3480, 4796, 5532, 5733, 7152, 7605, 8128, 9680, 11050, 12402, 15776, 33468, 36720, 37064, 38408, 43584, 50960, 55216, 63708, 70364, 83772, 92280, 106700, 114840, 116288, 149400, 163800, 166617, 167580

Offset: 1

S. I. Dimitrov, Jul 02 2025

The numbers x, y, z and t form a sigma-quadratic quadruple. See Dimitrov link.

			(3480, 3672, 4296, 8520) is such a quadruple because sigma(3480)^2 = sigma(3672)^2 = sigma(4296)^2 = 3480^2 + 3672^2 + 4296^2 + 8520^2.

David A. Corneth, Table of n, a(n) for n = 1..97
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David A. Corneth, PARI program
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A000414, A385356, A385397.

isok(x) = my(s=sigma(x), vi=select(t->(t>=x), invsigma(s))); for (i=1, #vi, for (j=1, #vi, for (k=1, #vi, if ((i==1) || (j==1) || (k==1), my(ss = s^2 - vi[i]^2 - vi[j]^2 - vi[k]^2); if (ss && issquare(ss), return(1)););););); \\ Michel Marcus, Jul 09 2025

PARI
```
\\ See Corneth link
```

Some missing terms added by Michel Marcus, Jul 09 2025

More terms from David A. Corneth, Jul 09 2025

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 3, 0, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1

Offset: 0

Felix Huber, Jul 22 2025

nonn

a(n) is the number of partitions of n into 4 nonzero squares < n/2.

			The a(51) = 1 multiset is [1, 9, 16, 25].
The a(52) = 3 multisets are [1, 1, 25, 25], [4, 16, 16, 16] and [9, 9, 9, 25].

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A000290, A000414, A000534, A025428, A062890, A385736.

# After Alois P. Heinz (A025428)
b:=proc(n,i,t)
    option remember;
    `if`(n=0,`if`(t=0,1,0),`if`(i<1 or t<1, 0, b(n,i-1,t)+`if`(i^2>n,0,b(n-i^2,i,t-1))))
    end:
A385860:=n->b(n,floor(sqrt((n-1)/2)),4):
seq(A385860(n),n=0..87);

a(n) <= A025428(n).

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

A215537 Lowest k such that k is representable as both the sum of n and of n+1 nonzero squares.

Original entry on oeis.org

25, 17, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Jon Perry, Aug 15 2012

nonn

			25 = 5^2 = 3^2 + 4^2
17 = 4^2 + 1^2 = 3^2 + 2^2 + 2^2
12 = 2^2 + 2^2 + 2^2 = 3^2 + 1^2 + 1^2 + 1^2
after this just add 1^2 to both sides.

Cf. A000290 (representable as sum of 1 square), A000404 (sum of 2 positive squares), A000408 (sum of 3 positive squares), A000414 (sum of 4 positive squares), A047700 (sum of 5 positive squares)

# true if a is representable as a sum of n squares, each square >= m^2.
isRepnSqrsMin := proc(a,n,m)
    local mpr ;
    if a < n*m^2 then
        return false;
    end if;
    if n = 1 then
        if a>= m^2 and issqr(a) then
            true;
        else
            false;
        end if;
    else
        for mpr from m to a do
            if a-mpr^2 < 1 then
                return false;
            elif procname(a-mpr^2,n-1,mpr) then
                return true;
            end if;
        end do:
    end if;
end proc:
# true if a is representable as a sum of n positive squares.
isRepnSqrs := proc(a,n)
    isRepnSqrsMin(a,n,1) ;
end proc:
A215537 := proc(n)
    local k;
    for k from 1 do
        if isRepnSqrs(k,n) and isRepnSqrs(k,n+1) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 11 2012

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A003386 Numbers that are the sum of 8 nonzero 8th powers.

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A097203 Number of 4-tuples (a,b,c,d) with 1 <= a <= b <= c <= d, a^2+b^2+c^2+d^2 = n and gcd(a,b,c,d) = 1.

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A145202 Primes of form 4*n^2 + 4*n + 653.

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A347803 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^4.

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A360530 a(n) is the smallest positive integer k such that n can be expressed as the arithmetic mean of k nonzero squares.

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A145125 Primes of the form 4n^2-4n+653.

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A385531 Numbers x such that there exist three integers 00 such that sigma(x)^2 = sigma(y)^2 = sigma(z)^2 = x^2 + y^2 + z^2 + t^2.

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A145202 Primes of form 4n^2 + 4n + 653.