A154777 -id:A154777 - cp's OEIS frontend

A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

5, 8, 13, 17, 20, 25, 29, 32, 37, 40, 41, 45, 52, 53, 61, 65, 68, 72, 73, 80, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 125, 128, 136, 137, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 197, 200, 205, 208, 212, 221, 225, 229, 232, 233

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and A024352 (difference of squares).

Also: Numbers of the form x^2+4y^2, where x and y are positive integers. Cf. A154777, A092572, A154778 for analogous sequences. - M. F. Hasler, Jan 24 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097269, A097270, A097271.

isA097268(n) = forstep( b=2,sqrtint(n-1),2, issquare(n-b^2) && return(1)) \\ M. F. Hasler, Jan 24 2009

A155716 Numbers of the form N = a^2 + 6b^2 for some positive integers a,b.

Original entry on oeis.org

7, 10, 15, 22, 25, 28, 31, 33, 40, 42, 49, 55, 58, 60, 63, 70, 73, 79, 87, 88, 90, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 145, 150, 151, 154, 159, 160, 166, 168, 175, 177, 186, 193, 196, 198, 199, 202, 214, 217, 220, 223, 225, 231, 232, 240

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A002481 (which allows for a and b to be zero).

Primes are in A033199. - Bernard Schott, Sep 20 2019

David A. Corneth, Table of n, a(n) for n = 1..17743 (terms <= 10^5)

Cf. A002481, A033199.

Cf. A000404, A154777, A092572, A097268, A154778, A155707-A155717, A155560-A155578.

With[{upto=240},Select[Union[#[[1]]^2+6#[[2]]^2&/@Tuples[ Range[Sqrt[ upto]], 2]],#<=upto&]] (* Harvey P. Dale, Aug 05 2016 *)

isA155716(n,/* optional 2nd arg allows us to get other sequences */c=6) = { for(b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,999, isA155716(n) & print1(n","))

upto(n) = my(res=List()); for(i=1,sqrtint(n),for(j=1, sqrtint((n - i^2) \ 6), listput(res, i^2 + 6*j^2))); listsort(res,1); res \\ David A. Corneth, Sep 18 2019

A155578 Intersection of A000404 and A155717: N = a^2 + b^2 = c^2 + 7*d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

8, 29, 32, 37, 53, 72, 109, 113, 116, 128, 137, 148, 149, 193, 197, 200, 212, 232, 233, 261, 277, 281, 288, 296, 317, 333, 337, 373, 389, 392, 400, 401, 421, 424, 436, 449, 452, 457, 464, 477, 512, 541, 548, 557, 569, 592, 596, 613, 617, 641, 648, 653, 673

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155568 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717.

isA155578(n,/* optional 2nd arg allows us to get other sequences */c=[7,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155578(n) & print1(n","))

from math import isqrt
def aupto(limit):
    cands = range(1, isqrt(limit)+1)
    left =  set(a**2 +   b**2 for a in cands for b in cands)
    right = set(c**2 + 7*d**2 for c in cands for d in cands)
    return sorted(k for k in left & right if k <= limit)
print(aupto(673)) # Michael S. Branicky, Aug 29 2021

A155717 Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.

Original entry on oeis.org

8, 11, 16, 23, 29, 32, 37, 43, 44, 53, 56, 64, 67, 71, 72, 77, 79, 88, 92, 99, 107, 109, 112, 113, 116, 121, 127, 128, 137, 144, 148, 149, 151, 161, 163, 172, 176, 179, 184, 191, 193, 197, 200, 203, 207, 211, 212, 224, 232, 233, 239, 253, 256, 259, 261, 263, 268

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A020670 (which allows for a and b to be zero).

If N=a^2+7*b^2 is a term then 7*N=(7*b)^2+7*a^2 is also a term. Conversely,if 7*N is a term then N is a term. Example: N=56; N/7=8 is a term, N*7=7^2+7*7^2 is a term. Sequences A154777, A092572 and A154778 have the same property with 7 replaced by prime numbers 2,3 and 5 respectively. - Jerzy R Borysowicz, May 22 2020

Cf. A000404, A154777, A092572, A097268, A154778, A155707-A155716, A155560-A155578.

Select[Range[300], Reduce[a>0 && b>0 && # == a^2 + 7b^2, {a, b}, Integers] =!= False&] (* Jean-François Alcover, Nov 17 2016 *)

isA155717(n,/* optional 2nd arg allows us to get other sequences */c=7) = { for(b=1,sqrtint((n-1)\c), issquare(n-c*b^2) & return(1))}
for( n=1,300, isA155717(n) & print1(n","))

def aupto(limit):
    cands = range(1, int(limit**.5)+2)
    nums = [a**2 + 7*b**2 for a in cands for b in cands]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(268)) # Michael S. Branicky, Aug 11 2021

A216451 Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>0, y>0.

Original entry on oeis.org

193, 337, 457, 673, 772, 1009, 1033, 1129, 1201, 1297, 1348, 1737, 1801, 1828, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2692, 2713, 2857, 3033, 3049, 3088, 3217, 3313, 3361, 3529, 3600, 3697, 3889, 4036, 4057, 4113, 4132, 4153, 4201, 4516, 4561, 4624, 4657

Offset: 1

V. Raman, Sep 07 2012

nonn

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+2*b^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+3*b^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+7*b^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. Also the power of 2 should not be 1, if it can be written in the form a^2+7*b^2.

V. Raman, Table of n, a(n) for n = 1..1000

Cf. A154777, A092572.

Intersection of A001481, A002479, A003136 and A020670, omitting squares. See also A216500. - N. J. A. Sloane, Sep 11 2012

nn = 4657; lim = Floor[Sqrt[nn]]; t1 = Select[Union[Flatten[Table[a^2 + b^2, {a, lim}, {b, lim}]]], # <= nn &]; t2 = Select[Union[Flatten[Table[a^2 + 2*b^2, {a, lim}, {b, lim/Sqrt[2]}]]], # <= nn &]; t3 = Select[Union[Flatten[Table[a^2 + 3*b^2, {a, lim}, {b, lim/Sqrt[3]}]]], # <= nn &]; t7 = Select[Union[Flatten[Table[a^2 + 7*b^2, {a, lim}, {b, lim/Sqrt[7]}]]], # <= nn &]; Intersection[t1, t2, t3, t7] (* T. D. Noe, Sep 08 2012 *)

Definition clarified by N. J. A. Sloane, Sep 11 2012

A216501 Let S_k = {x^2+k*y^2: x,y positive integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?

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

Offset: 1

V. Raman, Sep 07 2012

nonn

"If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor P (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.

Cf. A000290, A001481, A002479, A003136, A020670, A216451, A216500, A154777, A092572.

for(n=1, 100, sol=0; for(x=1, 100, if(issquare(n-x*x)&&n-x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-2*x*x)&&n-2*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-3*x*x)&&n-3*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-7*x*x)&&n-7*x*x>0, sol++; break)); print1(sol", ")) /* V. Raman, Oct 16 2012 */

a(n) = 0 for almost all n. - Charles R Greathouse IV, Sep 14 2012

Edited by N. J. A. Sloane, Sep 11 2012

A216503 a(n) = number of positive integers k such that n = x^2 + k*y^2 has a solution with x>0, y>0.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 3, 3, 4, 6, 5, 5, 5, 5, 5, 4, 6, 7, 5, 5, 6, 8, 5, 6, 7, 7, 7, 5, 9, 10, 6, 6, 9, 11, 6, 8, 9, 10, 7, 6, 10, 11, 7, 7, 11, 11, 8, 7, 10, 11, 9, 8, 9, 14, 7, 9, 10, 11, 8, 10, 13, 11, 9, 8, 15, 17, 9, 9, 14, 11, 8, 11, 12, 14

Offset: 1

V. Raman, Sep 07 2012

nonn

			a(17) = 6 because
17 = 4^2 + 1*1^2.
17 = 3^2 + 2*2^2.
17 = 1^2 + 4*2^2.
17 = 3^2 + 8*1^2.
17 = 2^2 + 13*1^2.
17 = 1^2 + 16*1^2.
Therefore there are 6 different values of k for which 17 can be written in the form a^2 + k*b^2.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001481, A154777, A092572.

Table[cnt = 0; Do[b = 1; found = False; While[q = n - k*b^2; ! found && q > 0, If[IntegerQ[Sqrt[q]], cnt++; found = True]; b++], {k, n}]; cnt, {n, 100}] (* T. D. Noe, Sep 11 2012 *)

for(n=1, 100, sol=0; for(k=1, n-1, for(x=1, n, if(issquare(n-k*x*x)&&n-k*x*x>0, sol++; break))); print1(sol", ")) /* V. Raman, Oct 16 2012 */

A216504 Number of values of k for which n can be written in the form a^2 + k*b^2, a >= 0, b >= 0, k > 0. a(n) = 0 if there are infinitely many such k.

Original entry on oeis.org

0, 2, 2, 0, 3, 3, 3, 5, 0, 4, 4, 5, 6, 4, 4, 0, 7, 6, 6, 7, 6, 6, 5, 8, 0, 6, 7, 8, 9, 6, 7, 10, 8, 8, 6, 0, 11, 7, 7, 11, 12, 7, 9, 11, 11, 8, 7, 11, 0, 9, 8, 13, 12, 10, 8, 12, 12, 10, 9, 11, 15, 8, 11, 0, 12, 9, 11, 15, 12, 10, 9, 17, 18, 10, 11, 16, 12, 9, 12, 15, 0, 12, 10, 14, 14, 11, 10, 17, 18, 13, 11, 15, 15, 12, 10, 17, 21, 12, 14, 0

Offset: 1

V. Raman, Sep 07 2012

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+2*b^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+3*b^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+7*b^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. Also the power of 2 should not be 1, if it can be written in the form a^2+7*b^2.
a(n) = 0 if and only if n is a square. - Charles R Greathouse IV, Sep 11 2012

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

Cf. A001481, A154777, A092572.

N:= 100: # for a(1)..a(N)
m:= floor(sqrt(N)):
V:= Vector(N,i->{}):
for a from 0 to m do
  for b from 1 to m do
    for k from 1 to floor((N-a^2)/b^2) do
      x:= a^2 + k*b^2;
      V[x]:= V[x] union {k};
od od od:
for i from 1 to N do
  if issqr(i) then V[i]:=0 else V[i]:= nops(V[i]) fi
od:
convert(V,list); # Robert Israel, Mar 06 2025

for(n=1, 100, sol=0; for(k=1, n, for(x=0, n, if(issquare(n-k*x*x)&&n-k*x*x>=0, sol++; break))); if(issquare(n), print1(0", "), print1(sol", "))) /* V. Raman, Oct 16 2012 */

A216505 Number of values of k for which n can be written in the form a^2+k*b^2, a > 0, b > 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 4, 5, 3, 3, 5, 6, 5, 5, 5, 5, 5, 4, 6, 8, 5, 5, 6, 8, 5, 6, 7, 7, 7, 5, 10, 10, 6, 6, 9, 11, 6, 8, 9, 10, 7, 6, 10, 12, 7, 7, 11, 11, 8, 7, 10, 11, 9, 8, 9, 14, 7, 9, 11, 11, 8, 10, 13, 11, 9, 8, 15, 17, 9, 9, 14, 11, 8, 11, 12

Offset: 1

V. Raman, Sep 07 2012

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001481, A154777, A092572.

Table[cnt = 0; If[IntegerQ[Sqrt[n]], cnt++]; Do[b = 1; found = False; While[q = n - k*b^2; ! found && q > 0, If[IntegerQ[Sqrt[q]], cnt++; found = True]; b++], {k, n}]; cnt, {n, 100}] (* T. D. Noe, Sep 11 2012 *)

for(n=1, 100, sol=0; for(k=0, n-1, for(x=1, n, if(issquare(n-k*x*x)&&n-k*x*x>0, sol++; break))); print1(sol", ")) /* V. Raman, Oct 16 2012 */

A216671 Let S_k = {x^2+k*y^2: x,y nonnegative integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 13 2012

nonn

"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.
Comment from N. J. A. Sloane, Sep 14 2012: S_1, S_2, S_3, S_7 are the first four quadratic forms with class number 1. (See Cox, for example.)

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989. - From N. J. A. Sloane, Sep 14 2012

Cf. A000290, A001481, A002479, A003136, A020670, A216451, A216500.

Cf. A154777, A092572.

for(n=1, 100, sol=0; for(x=0, 100, if(issquare(n-x*x)&&n-x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-2*x*x)&&n-2*x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-3*x*x)&&n-3*x*x>=0, sol++; break)); for(x=0, 100, if(issquare(n-7*x*x)&&n-7*x*x>=0, sol++; break)); print1(sol", ")) /* V. Raman, Oct 16 2012 */

The fraction of terms with a(n)>0 goes to zero as n increases. - Charles R Greathouse IV, Sep 11 2012

Edited by N. J. A. Sloane, Sep 14 2012

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A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

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A155716 Numbers of the form N = a^2 + 6b^2 for some positive integers a,b.

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A155578 Intersection of A000404 and A155717: N = a^2 + b^2 = c^2 + 7*d^2 for some positive integers a,b,c,d.

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A155717 Numbers of the form N = a^2 + 7b^2 for some positive integers a,b.

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A216451 Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>0, y>0.

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A216501 Let S_k = {x^2+k*y^2: x,y positive integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?

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A216503 a(n) = number of positive integers k such that n = x^2 + k*y^2 has a solution with x>0, y>0.

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A216504 Number of values of k for which n can be written in the form a^2 + k*b^2, a >= 0, b >= 0, k > 0. a(n) = 0 if there are infinitely many such k.

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