A216679 -id:A216679 - cp's OEIS frontend

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.

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.

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

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.

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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.

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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.

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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.

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cp's OEIS Frontend

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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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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Author

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Programs

Maple

Mathematica

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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Author

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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.

Views

Author

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Comments

Crossrefs

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.

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.