A025426 -id:A025426 - cp's OEIS frontend

A063665 Number of ways 1/n can be written as 1/x^2 + 1/y^2 with y >= x >= 1.

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

Offset: 1

Henry Bottomley, Jul 25 2001

nonn

Number of ordered pairs (x,y), with n = (x^2)(y^2)/(x^2 + y^2) and y >= x > 0. - Antti Karttunen, Nov 07 2018

			a(90)=1 since 1/90 = 1/10^2 + 1/30^2
a(98)=2 since 1/98 = 1/10^2 + 1/70^2 = 1/14^2 + 1/14^2.
a(14400) = 3 since 1/14400 = 1/130^2 + 1/312^2 = 1/136^2 + 1/255^2 = 1/150^2 + 1/200^2. - _Antti Karttunen_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000161, A025426, A018892, A063664.

A063665(n) = { my(s=0); for(x=1,n,for(y=x,n,if((n*(x*x+y*y)) == (x*x*y*y), s++))); (s); }; \\ Antti Karttunen, Nov 07 2018

A063665(n) = { my(s=0,y); for(x=sqrtint(n),n,my(x2=x*x); if((x2>n)&&issquare((n*x2)/(x2-n),&y)&&(1==denominator(y))&&(y>=x),s++)); (s); }; \\ Antti Karttunen, Nov 07 2018

Definition clarified by Antti Karttunen, Nov 07 2018

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

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

A216280 Number of nonnegative solutions to the equation x^4 + y^4 = n.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

V. Raman, Sep 03 2012

nonn

The first n with a(n) > 1 is 635318657 = 41 * 113 * 241 * 569, with a(635318657) = 2. Izadi, Khoshnam, & Nabardi show that for any n with a(n) > 1, the elliptic curve y^2 = x^3 - nx has rank at least 3. According to gp, y^2 = x^3 - 635318657x has analytic rank 4 (and first nonzero derivative around 35741.7839). - Charles R Greathouse IV, Jan 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16561
F. A. Izadi, F. Khoshnam, K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥4 (2012). arXiv:1202.5676 [math.NT], 2012.
F. A. Izadi, F. Khoshnam, K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥ 4, Math. J. Okayama Univ. 56 (2014), 51-63.

Cf. A025455, A025446, A000161, A025426, A216284.

Cf. A004831 (positions of nonzero terms).

Reap[For[n = 1, n <= 1000, n++, r = Reduce[0 <= x <= y && x^4 + y^4 == n, {x, y}, Integers]; sols = Which[r === False, 0, r[[0]] == And, 1, r[[0]] == Or, Length[r], True, Print[n, " ", r]]; If[sols != 0, Print[n, " ", sols, " ", r]]; Sow[sols]]][[2, 1]] (* Jean-François Alcover, Feb 22 2019 *)

a(n)=my(t=thue(thueinit('x^4+1,1),n)); sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1]) \\ Charles R Greathouse IV, Jan 12 2017

first(n)=my(T=thueinit('x^4+1,1),v=vector(n),t); for(k=1,n, t=thue(T,k); v[k]=sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1])); v \\ Charles R Greathouse IV, Jan 12 2017

Offset added by Charles R Greathouse IV, Jan 12 2017

A274567 Least number k such that k^2-1 is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

3, 81, 51, 291, 1251, 339, 62499, 1971, 5201, 5001, 175781251, 7299

Offset: 1

Altug Alkan, Jun 28 2016

a(11) > 25*10^5 if it exists. - Chai Wah Wu, Jul 23 2020
From David A. Corneth, Jul 23 2020: (Start)
a(13) <= 17578125001, a(17) <= 610351562499. (End)

			a(2) = 81 because 81^2 - 1 = 28^2 + 76^2 = 44^2 + 68^2.

Cf. A000404, A005563, A016032, A025426, A050795, A050797, A054994.

a(10) from Chai Wah Wu, Jul 22 2020

A336542 Primitive integers for the number of ways k to write as a sum of two squares.

Original entry on oeis.org

1, 2, 5, 10, 25, 50, 65, 125, 130, 250, 325, 625, 650, 1105, 1250, 1625, 2210, 3125, 3250, 4225, 5525, 6250, 8125, 8450, 11050, 15625, 16250, 21125, 27625, 31250, 32045, 40625, 42250, 55250, 64090, 71825, 78125, 81250, 105625, 138125, 143650, 156250, 160225, 203125, 211250

Offset: 1

David A. Corneth, Jul 24 2020

nonn

The number of ways to write k as a number of two squares only depends on the parity of the multiplicity of 2, the parity of the multiplicity of a prime of the form 4*m + 3 and the multiplicity of a prime of the form 4*m+1 (See A025426). Terms in this sequence have no prime factors of the form 4*m + 3.

			650 = 2*5*13 is in the sequence as its prime factors are 2 or of the form 4*m + 1. It's the least positive integer of the form 2*p*q where p and q are distinct and each of the form 4*m+1.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A025426, A054994, A274567.

A211339 Number of integer pairs (x,y) such that 1 < x <= y <= n and x^2 + y^2 <= n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 25, 25, 25, 26, 26, 26, 26, 27, 28, 29

Offset: 1

Clark Kimberling, Apr 08 2012

nonn

Partial sums of A025426.

For a guide to related sequences, see A211266.

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x^2 + y^2, {x, a, b - 1}, {y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
TableForm[Table[c[n, k], {n, 1, 10}, {k, 1, 2 n}]]
Table[c[n, n], {n, 1, z1}]   (* A025426 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]  (* A211339 *)

a(n) = -1/2(-1 + floor(sqrt(n/2)))(floor(sqrt(n/2))) + Sum_{k=1..floor(sqrt(n/2))} floor(sqrt(n - k^2)). - Nicholas Stearns, Apr 03 2017

A217463 a(n) is the sum of total number of positive integer solutions to each of a^2 + b^2 = n, a^2 + 2b^2 = n, a^2 + 3b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).

Original entry on oeis.org

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

Offset: 1

V. Raman, Oct 04 2012

nonn

Note: For the equation a^2 + b^2 = n, if there are two solutions (a,b) and (b,a), then they will be counted only once.
The sequences A216501 and A216671 give how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to.
1, 2, 3, 7 are the first four numbers, with the class number 1.
"If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor of C raised to an odd power is of the form c^2 + kd^2 for some integers c & d."
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
A number can be written as a^2 + b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
A number can be written as a^2 + 2b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
A number can be written as a^2 + 3b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
A number can be written as a^2 + 7b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power, and the exponent of 2 is not 1.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Cf. A216501, A216671.
Cf. A217869 (related sequence of this when the order does matter for the equation a^2 + b^2 = n).
Cf. A216501 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a > 0, b > 0).
Cf. A216671 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a >= 0, b >= 0).
Cf. A025426 (number of solutions to n = a^2+b^2 (when the solutions (a, b) and (b, a) are being counted as the same) with a > 0, b > 0).
Cf. A216278 (number of solutions to n = a^2+2*b^2 with a > 0, b > 0).
Cf. A092573 (number of solutions to n = a^2+3*b^2 with a > 0, b > 0).
Cf. A216511 (number of solutions to n = a^2+7*b^2 with a > 0, b > 0).

for(n=1,100,sol=0;for(x=1,100,if(issquare(n-x*x)&&n-x*x>0&&x*x<=n-x*x,sol++);if(issquare(n-2*x*x)&&n-2*x*x>0,sol++);if(issquare(n-3*x*x)&&n-3*x*x>0,sol++);if(issquare(n-7*x*x)&&n-7*x*x>0,sol++));printf(sol","))

A273279 Least perfect power that is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

8, 125, 3125, 4225, 1953125, 48828125, 105625, 274625, 762939453125, 2640625, 476837158203125, 17850625, 1221025, 34328125, 186264514923095703125, 1650390625, 446265625, 1160290625, 41259765625, 4291015625, 45474735088646411895751953125, 30525625

Offset: 1

Altug Alkan, May 19 2016

nonn

Least m^k that is the sum of two nonzero squares in exactly n ways where m > 0 and k >= 2.
Terms of this sequence are 2^3, 5^3, 5^5, 65^2, 5^10, 5^11, 325^2, 65^3, ...
Prime powers that are listed in this sequence are 2^3, 5^3, 5^5, 5^10, 5^11, ...

			8 is a term because 8 = 2^3 = 2^2 + 2^2.
125 is a term because 125 = 5^3 = 2^2 + 11^2 = 5^2 + 10^2.
3125 is a term because 3125 = 5^5 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A001597, A006339, A016032, A025426, A266927, A273238.

p = Select[Prime@ Range@ 90, Mod[#, 4] == 1 &]; f[w_] := Times @@ (Take[p, Length@w]^Reverse[w]); c[w_] := Floor[(1/2) Times @@ (w+1)];r[w_] := Block[{v, k = If[Length@w == 1, 1,2]}, While[(v = cn[k w]) < trg, k++]; If[v == trg, b = Min[b, f[k*w]]]; If[cn[w] < trg, r[Append[w, 1]]; v=w; v[[-1]]++; r[v]]]; a[1]=8; a[n_] := (b=Infinity; trg = n; r[{2}]; r[{1, 1}]; b); Array[a, 50] (* Giovanni Resta, May 19 2016 *)

a(9)-a(22) from Giovanni Resta, May 19 2016

cp's OEIS Frontend

A063665 Number of ways 1/n can be written as 1/x^2 + 1/y^2 with y >= x >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

A216280 Number of nonnegative solutions to the equation x^4 + y^4 = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A274567 Least number k such that k^2-1 is the sum of two nonzero squares in exactly n ways.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A336542 Primitive integers for the number of ways k to write as a sum of two squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A211339 Number of integer pairs (x,y) such that 1 < x <= y <= n and x^2 + y^2 <= n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A217463 a(n) is the sum of total number of positive integer solutions to each of a^2 + b^2 = n, a^2 + 2*b^2 = n, a^2 + 3*b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).

A217463 a(n) is the sum of total number of positive integer solutions to each of a^2 + b^2 = n, a^2 + 2b^2 = n, a^2 + 3b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).