A092573 -id:A092573 - cp's OEIS frontend

A025428 Number of partitions of n into 4 nonzero squares.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 4, 1, 3, 2, 1, 4, 1, 1, 3, 3, 2, 2, 4, 2, 1, 3, 2, 3, 4, 2, 3, 3, 1, 2, 5, 2, 4, 3, 2, 4, 1, 1, 6, 4, 3, 4, 2, 3, 0, 4, 4, 3, 5, 1, 5, 5, 1, 4, 5, 2

Offset: 0

David W. Wilson

Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015

Cf. A000414, A000534, A025357-A025375, A216374, A025416 (greedy inverse).

Column k=4 of A243148.

A025428 := proc(n)
    local a,i,j,k,lsq ;
    a := 0 ;
    for i from 1 do
        if 4*i^2 > n then
            return a;
        end if;
        for j from i do
            if i^2+3*j^2 > n then
                break;
            end if;
            for k from j do
                if i^2+j^2+2*k^2 > n then
                    break;
                end if;
                lsq := n-i^2-j^2-k^2 ;
                if lsq >= k^2 and issqr(lsq) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc:
seq(A025428(n),n=1..40) ; # R. J. Mathar, Jun 15 2018
# second Maple program:
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:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 14 2019

nn = 100; lim = Sqrt[nn]; t = Table[0, {nn}]; Do[n = a^2 + b^2 + c^2 + d^2; If[n <= nn, t[[n]]++], {a, lim}, {b, a, lim}, {c, b, lim}, {d, c, lim}]; t (* T. D. Noe, Sep 28 2012 *)
f[n_] := Length@ IntegerPartitions[n, {4}, Range[ Floor[ Sqrt[n - 1]]]^2]; Array[f, 105] (* Robert G. Wilson v, Sep 28 2012 *)

A025428(n)=sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,a^2+b^2+c^2+d^2==n))))

A025428(n)=sum(a=1,sqrtint(max(n-3,0)), sum(b=1,min(sqrtint(n-a^2-2),a), sum(c=1,min(sqrtint(n-a^2-b^2-1),b),issquare(n-a^2-b^2-c^2,&d) & d <= c )))

A025428(n)=sum(a=sqrtint(max(n,4)\4),sqrtint(max(n-3,0)), sum(b=sqrtint((n-a^2)\3-1)+1,min(sqrtint(n-a^2-2),a), sum(c=sqrtint((t=n-a^2-b^2)\2-1)+1, min(sqrtint(t-1),b), issquare(t-c^2) ))) \\ - M. F. Hasler, Sep 17 2012
for(n=1,100,print1(A025428(n),","))

T(n)={a=matrix(n,4,i,j,0);for(d=1,sqrtint(n),forstep(i=n,d*d+1,-1,for(j=2,4,a[i,j]+=sum(k=1,j,if(k0,a[i-k*d*d,j-k],if(k==j&&i-k*d*d==0,1)))));a[d*d,1]=1);for(i=1,n,print(i" "a[i,4]))} /* Robert Gerbicz, Sep 28 2012 */

For n>0, a(n) = ( A063730(n) + 6*A213024(n) + 3*A063725(n/2) + 8*A092573(n) + 6*A010052(n/4) ) / 24. - Max Alekseyev, Sep 30 2012
a(n) = ( A000118(n) - 4*A005875(n) - 6*A004018(n) - 12*A000122(n) - 15*A000007(n) + 12*A014455(n) - 24*A033715(n) - 12*A000122(n/2) + 12*A004018(n/2) + 32*A033716(n) - 32*A000122(n/3) + 48*A000122(n/4) ) / 384. - Max Alekseyev, Sep 30 2012
a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(n-i-j-k). - Wesley Ivan Hurt, Apr 19 2019

Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012

A092572 Numbers of the form x^2 + 3y^2 where x and y are positive integers.

Original entry on oeis.org

4, 7, 12, 13, 16, 19, 21, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 76, 79, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Superset of primes of the form 6n+1 (A002476).
It seems that all integer solutions of ((a+b)^3 - (a-b)^3) / (2*b) = c^3 have c = x^2 + 3*y^2. - Juergen Buchmueller (pullmoll(AT)t-online.de), Apr 04 2008
To prove the case of cubes in Fermat's last theorem, Euler considered numbers of the form a^2 + 3b^2. In the equation x^3 + y^3 = z^3, Euler specified that x = a - b and y = a + b. - Alonso del Arte, Jul 19 2012
All terms == 0,1,3,4, or 7 (mod 9). - Robert Israel, Apr 03 2017

			7 is of the specified form, since 2^2 + 3 * 1^2 = 7.
So is 12, since 3^2 + 3 * 1^2 = 12, and 13, with 1^2 + 3 * 2^2 = 13.

Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag (1979): 4.

Robert Israel, Table of n, a(n) for n = 1..10000
E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Cf. A002476, A092573, A092575, A158937 (similar definition but with duplicates left in).

N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2 + 3*y^2, x = 1 .. floor(sqrt(N - 3*y^2))),
  y=1..floor(sqrt(N/3-1)))}:
sort(convert(S,list)); # Robert Israel, Apr 03 2017

Union[Flatten[Table[a^2 + 3b^2, {a, 20}, {b, Ceiling[Sqrt[(400 - a^2)/3]]}]]] (* Alonso del Arte, Jul 19 2012 *)

A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Records occur at 1, 9, 81, 297, 891, 1683, 5049, 15147, 31977, ... - Antti Karttunen, Aug 23 2017

			For n = 9, there are two solutions: 9 = 9^2 + 2*(0^2) = 1^2 + 2*(2^2), thus a(9) = 2.
For n = 81, there are three solutions: 81 = 9^2 + 2*(0^2) = 3^2 + 2*(6^2) = 7^2 + 2*(4^2), thus a(81) = 3.
For n = 65536, there is one solution: 65536 = 256^2 + 2*(0^2) = 65536 + 0, thus a(65536) = 1.
For n = 65537, there is one solution: 65537 = 255^2 + 2*(16^2) = 65205 + 512, thus a(65537) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A092573, A119395, A000161, A025426, A216283.

Cf. A002479 (positions of nonzeros), A097700 (of zeros).

r[n_] := Reduce[x >= 0 && y >= 0 && x^2 + 2 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2017 *)

(define (A216282 n) (cond ((< n 2) 1) (else (let loop ((k (- (A000196 n) (modulo (- n (A000196 n)) 2))) (s 0)) (if (< k 0) s (let ((x (/ (- n (* k k)) 2))) (loop (- k 2) (+ s (A010052 x))))))))) ;; Antti Karttunen, Aug 23 2017

Examples from Antti Karttunen, Aug 23 2017

A374158 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 3*y^2 = k.

Original entry on oeis.org

0, 4, 91, 28, 196, 31213, 364, 9604, 53599, 2548, 470596

Offset: 0

Seiichi Manyama, Jun 29 2024

a(n) is the smallest nonnegative k such that A092573(k) = n.
a(11) <= 3672178237.
a(12) = 6916.
a(13) = 33124.
a(14) = 29059303.
a(15) = 124852.
a(16) = 1983163.
a(18) = 48412.
a(20) = 18384457.
a(21) = 6117748.
a(22) = 1623076.
a(24) = 214396.
a(27) = 629356.
a(28) = 900838393.
a(31) = 79530724.
a(32) = 85276009.
a(37) = 274299844.
a(42) = 116237212.
a(60) = 73537828.
a(67) = 585930436.
From Chai Wah Wu, Jun 29-30 2024: (Start)
a(30) = 2372188.
a(36) = 1500772.
a(40) = 11957764.
a(45) = 30838444.
a(48) = 7932652.
a(54) = 19510036.
a(72) = 55528564.
(End)

			   n | a(n)
-----+---------------------------
   1 |      4 = 2^2.
   2 |     91 = 7 * 13.
   3 |     28 = 2^2 * 7.
   4 |    196 = 2^2 * 7^2.
   5 |  31213 = 7^4 * 13.
   6 |    364 = 2^2 * 7 * 13.
   7 |   9604 = 2^2 * 7^4.
   8 |  53599 = 7 * 13 * 19 * 31.
   9 |   2548 = 2^2 * 7^2 * 13.
  10 | 470596 = 2^2 * 7^6.

Cf. A328151, A343105, A374159, A374160, A374161.

Cf. A002476, A092573.

from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374158(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x**2+3*y**2-m) if d[0]>0 and d[1]>0)==n) # Chai Wah Wu, Jun 29 2024

A092575 Number of representations of n of the form x^2 + 3y^2 with (x,y)=1.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Cf. A002476, A092572, A092573, A092574.

N:= 200: # to get a(1)..a(N)
V:= Vector(N):
for y from 1 to floor(sqrt(N/3-1)) do
  for x from 1 to floor(sqrt(N-3*y^2)) do
    if igcd(x,y) = 1 then V[x^2 + 3*y^2]:= V[x^2+3*y^2]+1
    fi
od od:
convert(V,list); # Robert Israel, Apr 03 2017

r[n_] := Reduce[ x > 0 && y > 0 && GCD[x, y] == 1 && n == x^2 + 3 y^2, {x, y}, Integers]; a[n_] := Which[ r[n] === False, 0, r[n][[0]] === And, 1, True, Length[r[n]]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Oct 31 2012 *)

A216283 Number of nonnegative solutions to the equation x^2+5*y^2 = n.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Records occur at 1, 9, 81, 189, 441, 1449, 3969, 12789, 13041, 30429, ... - Antti Karttunen, Aug 23 2017

			For n = 9, there are two solutions: 9 = 3^2 + 5*(0^2) = 2^2 + 5*(1^2), thus a(9) = 2.
For n = 81, there are three solutions: 81  = 9^2 + 5*(0^2) = 6^2 + 5*(3^2) = 1^2 + 5*(4^2), thus a(81) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033718 (all solutions x^2+5*y^2 = n).
Cf. A092573, A119395, A000161, A025426, A216282.
Cf. A020669 (positions of nonzeros).

N=666;  x='x+O('x^N);
T(x)=sum(n=0,ceil(sqrt(N)),x^(n*n));
Vec(T(x)*T(x^5))
/* Joerg Arndt, Sep 21 2012 */

(define (A216283 n) (cond ((< n 2) 1) (else (let loop ((k (A000196 n)) (s 0)) (if (< k 0) s (let ((x (- n (* k k)))) (loop (- k 1) (+ s (if (zero? (modulo x 5)) (A010052 (/ x 5)) 0))))))))) ;; Antti Karttunen, Aug 23 2017

G.f. T(x) * T(x^5) where T(x) = sum(n>=0, x^(n^2) ). - Joerg Arndt, Sep 21 2012

Examples from Antti Karttunen, Aug 23 2017

A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

Original entry on oeis.org

4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259, 268

Offset: 1

Eric W. Weisstein, Feb 28 2004

nonn

Superset of primes of the form 6n+1 (A002476).

For all proper solutions with nonnegative x and y see A244819. - Wolfdieter Lang, Mar 02 2021

E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Eulers 6n Plus 1 Theorem

Cf. A002476, A092572, A092573, A092575, A244819.

A216278 Number of solutions to the equation x^2+2y^2 = n with x and y > 0.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Cf. A092573, A119395, A000161, A025426.

r[n_] := Reduce[x > 0 && y > 0 && x^2 + 2 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2017 *)

A216279 Number of solutions to the equation x^2+5y^2 = n with x and y > 0.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A092573, A119395, A000161, A025426.

a(n)=sum(k=1,sqrtint((n-1)\5), issquare(n-5*k^2)) \\ Charles R Greathouse IV, Jun 06 2016

list(lim)=my(v=vector(lim\1),t); for(y=1,sqrtint((#v-1)\5), t=5*y^2; for(x=1,sqrtint(#v-t), v[x^2+t]++)); v \\ Charles R Greathouse IV, Jun 06 2016

A374017 Number of solutions to x^2 + 11*y^2 = n in positive integers x and y.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jun 29 2024

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

Cf. A092573, A216279, A216511.

Cf. A374160.

a(n) = if(n==0, 0, sum(k=1, sqrtint((n-1)\11), issquare(n-11*k^2)));

cp's OEIS Frontend

A025428 Number of partitions of n into 4 nonzero squares.

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A092572 Numbers of the form x^2 + 3y^2 where x and y are positive integers.

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A216282 Number of nonnegative solutions to the equation x^2 + 2*y^2 = n.

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A374158 a(n) is the smallest nonnegative integer k where exactly n pairs of positive integers (x, y) exist such that x^2 + 3*y^2 = k.

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A092575 Number of representations of n of the form x^2 + 3y^2 with (x,y)=1.

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Maple

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A216283 Number of nonnegative solutions to the equation x^2+5*y^2 = n.

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Programs

PARI

Scheme

Formula

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A092574 Positive integers that can be represented in the form x^2 + 3y^2 with (x,y) = 1 and x and y positive.

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A216278 Number of solutions to the equation x^2+2y^2 = n with x and y > 0.