A216500 -id:A216500 - cp's OEIS frontend

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.

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

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

A216679 Numbers which can be written neither as a^2+b^2, nor as a^2+2b^2, nor as a^2+3b^2, nor as a^2+7*b^2, with a >= 0 and b >= 0.

Original entry on oeis.org

14, 15, 30, 35, 42, 46, 47, 55, 60, 62, 69, 70, 78, 87, 94, 95, 105, 110, 115, 119, 120, 126, 135, 138, 140, 141, 142, 143, 154, 155, 158, 159, 165, 167, 168, 174, 182, 186, 188, 190, 195, 206, 210, 213, 215, 220, 222, 230, 231, 235, 238, 240, 248, 254, 255, 266, 270, 276, 280, 282, 285, 286, 287, 295, 299

Offset: 1

V. Raman, Sep 13 2012

nonn

If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^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.

Cf. A216451, A216500.

A216680 Numbers which can be written neither as a^2+b^2, nor as a^2+2b^2, nor as a^2+3b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

Original entry on oeis.org

1, 14, 15, 30, 35, 42, 46, 47, 55, 60, 62, 69, 70, 78, 87, 94, 95, 105, 110, 115, 119, 120, 126, 135, 138, 140, 141, 142, 143, 154, 155, 158, 159, 165, 167, 168, 174, 182, 186, 188, 190, 195, 206, 210, 213, 215, 220, 222, 230, 231, 235, 238, 240, 248, 254, 255, 266, 270, 276, 280, 282, 285, 286, 287, 295, 299

Offset: 1

V. Raman, Sep 13 2012

nonn

If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^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.
Essentially the same as A216679. - R. J. Mathar, Sep 16 2012

Cf. A216451, A216500.

A216682 Perfect squares which can be written in all the four forms a^2+b^2, a^2+2b^2, a^2+3b^2 and a^2+7*b^2, with a > 0 and b > 0.

Original entry on oeis.org

3600, 4624, 12100, 12321, 14400, 18496, 20449, 24336, 26896, 30276, 32400, 37249, 41616, 46225, 48400, 49284, 51076, 57600, 73984, 75076, 81796, 85264, 90000, 97344, 101124, 106929, 107584, 108900, 110889, 112225, 113569, 115600, 121104, 126736, 129600, 139876, 144400, 148225, 148996, 150544, 165649, 166464, 176400, 184041, 184900, 193600, 197136

Offset: 1

V. Raman, Sep 13 2012

nonn

If a composite number C, say, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^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.

Cf. A216451, A216500.

A216828 Numbers whose squares can be written in all the four forms a^2 + b^2, a^2 + 2b^2, a^2 + 3b^2 and a^2 + 7*b^2, with a > 0 and b > 0.

Original entry on oeis.org

60, 68, 110, 111, 120, 136, 143, 156, 164, 174, 180, 193, 204, 215, 220, 222, 226, 240, 272, 274, 286, 292, 300, 312, 318, 327, 328, 330, 333, 335, 337, 340, 348, 356, 360, 374, 380, 385, 386, 388, 407, 408, 420, 429, 430, 440, 444, 452, 457, 466, 468, 476, 480, 492, 522, 540, 544, 548, 550, 551, 555, 559, 562, 572, 579, 584

Offset: 1

V. Raman, Sep 17 2012

nonn

If a composite number C can be written in the form C = a^2 + k*b^2, for some integers a and b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2 + k*d^2, for some integers c and 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.

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

Cf. A216451, A216500, A216501, A216671, A216679, A216680, A216682.

filter:= proc(n) local L,x,y;
  select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+y^2)]) <> []
  and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+2*y^2)]) <> []
  and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+3*y^2)]) <> []
  and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+7*y^2)]) <> []
end proc:
select(filter, [$1..1000]); # Robert Israel, May 03 2018

okQ[n_] := Module[{x, y}, AllTrue[{1, 2, 3, 7}, Solve[x > 0 && y > 0 && n^2 == x^2 + #*y^2, {x, y}, Integers] =!= {}&]];
Select[Range[1000], okQ] (* Jean-François Alcover, May 23 2023 *)

A216408 Perfect squares which can be written neither as a^2+b^2, nor as a^2+2b^2, nor as a^2+3b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

Original entry on oeis.org

1, 2209, 27889, 96721, 146689, 229441, 253009, 418609, 516961, 703921, 786769, 966289, 1324801, 1495729, 1739761, 2211169, 2283121, 2430481, 3323329, 3411409, 4255969, 4879681, 5527201, 5755201, 7091569, 7219969, 8427409, 8994001, 9138529, 10029889, 10182481, 11282881, 11607649, 12439729, 13476241, 14922769, 15295921

Offset: 1

V. Raman, Sep 17 2012

nonn

If a composite number C, in case, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^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.

Cf. A216451, A216500, A216501, A216671, A216679, A216680, A216682

A216827 Numbers whose squares can be written neither as a^2 + b^2, nor as a^2 + 2b^2, nor as a^2 + 3b^2, nor as a^2 + 7*b^2, with a > 0 and b > 0.

Original entry on oeis.org

1, 47, 167, 311, 383, 479, 503, 647, 719, 839, 887, 983, 1151, 1223, 1319, 1487, 1511, 1559, 1823, 1847, 2063, 2209, 2351, 2399, 2663, 2687, 2903, 2999, 3023, 3167, 3191, 3359, 3407, 3527, 3671, 3863, 3911, 4007, 4079, 4583, 4679, 4703, 4751, 4871, 4919, 5039, 5087, 5351, 5519, 5591, 5711, 5879, 5927

Offset: 1

V. Raman, Sep 17 2012

nonn

If a composite number C can be written in the form C = a^2+k*b^2, for some integers a and b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c and 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.

Cf. A216451, A216500, A216501, A216671, A216679, A216680, A216682.

cp's OEIS Frontend

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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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?

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A216679 Numbers which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a >= 0 and b >= 0.

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A216680 Numbers which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

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A216682 Perfect squares which can be written in all the four forms a^2+b^2, a^2+2*b^2, a^2+3*b^2 and a^2+7*b^2, with a > 0 and b > 0.

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A216828 Numbers whose squares can be written in all the four forms a^2 + b^2, a^2 + 2*b^2, a^2 + 3*b^2 and a^2 + 7*b^2, with a > 0 and b > 0.

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A216408 Perfect squares which can be written neither as a^2+b^2, nor as a^2+2*b^2, nor as a^2+3*b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

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A216827 Numbers whose squares can be written neither as a^2 + b^2, nor as a^2 + 2*b^2, nor as a^2 + 3*b^2, nor as a^2 + 7*b^2, with a > 0 and b > 0.

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A216679 Numbers which can be written neither as a^2+b^2, nor as a^2+2b^2, nor as a^2+3b^2, nor as a^2+7*b^2, with a >= 0 and b >= 0.

A216680 Numbers which can be written neither as a^2+b^2, nor as a^2+2b^2, nor as a^2+3b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

A216682 Perfect squares which can be written in all the four forms a^2+b^2, a^2+2b^2, a^2+3b^2 and a^2+7*b^2, with a > 0 and b > 0.

A216828 Numbers whose squares can be written in all the four forms a^2 + b^2, a^2 + 2b^2, a^2 + 3b^2 and a^2 + 7*b^2, with a > 0 and b > 0.

A216408 Perfect squares which can be written neither as a^2+b^2, nor as a^2+2b^2, nor as a^2+3b^2, nor as a^2+7*b^2, with a > 0 and b > 0.

A216827 Numbers whose squares can be written neither as a^2 + b^2, nor as a^2 + 2b^2, nor as a^2 + 3b^2, nor as a^2 + 7*b^2, with a > 0 and b > 0.