cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A066647 Squares of the form a^2 + b^3 with a, b > 0.

Original entry on oeis.org

9, 36, 100, 196, 225, 289, 441, 576, 784, 841, 1225, 1296, 1764, 1849, 2025, 2304, 3025, 3249, 3600, 3844, 3969, 4356, 5776, 6084, 6400, 6561, 8100, 8281, 9801, 11025, 12544, 13924, 14161, 14400, 15129, 16129, 16641, 17424, 18496, 19600, 19881, 20449, 21609, 23409
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 17 2001

Keywords

Examples

			29^2 = a(9) = 841 = 625 + 216 = 25^2 + 6^3.

Links

Crossrefs

Cf. A055394, A066648, A070745.

Programs

  • Mathematica
    q[n_] := Length[Reduce[a^2 + b^3 == n && a > 0 && b > 0, {a, b}, Integers]] > 0; Select[Range[140]^2, q] (* Amiram Eldar, Mar 20 2025 *)

Formula

a(n) = A070745(n)^2. - Amiram Eldar, Mar 20 2025

A293283 Numbers n such that n^2 = a^2 + b^5 for positive integers a b and n.

Original entry on oeis.org

6, 9, 18, 40, 42, 68, 75, 90, 99, 105, 122, 126, 130, 174, 192, 196, 225, 251, 257, 288, 315, 325, 330, 350, 405, 490, 499, 504, 516, 528, 546, 550, 576, 614, 651, 665, 684, 726, 735, 744, 849, 882, 900, 920, 936, 974, 1025, 1032, 1036, 1107, 1140, 1183, 1200
Offset: 1

Views

Author

XU Pingya, Oct 04 2017

Keywords

Comments

For n > 0, k = (n + 1)(2n + 1)^2 is a term in this sequence, because k^2 = (n * (2n + 1)^2)^2 + (2n + 1)^5. Examples: 18, 75, 196, 405, 726, 1183.
When z^2 = x^2 + y^2 (i.e., z = A009003(n)), (z * y^4)^2 = (x * y^4)^2 + (y^2)^5. Thus z * y^4 is a term in this sequence. For example, 1200. More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 1) * z^(5k - 5) is in this sequence.
When z^2 = x^2 + y^3 (i.e., z = A070745(n)), (z * y)^2 = (x * y)^2 + y^5. Thus z * y is in this sequence. E.g. 6, 18, 40, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 4) * z^(5k - 4) is in this sequence.
When z^2 = x^2 + y^4 (i.e., z = A271576(n)), (z * y^3)^2 = (x * y^3)^2 + (y^2)^5. Thus z * y^3 is also in this sequence. E.g. 40, 405, 1107, ... . More generally, for positive integer i, j and k, x^(5i - 5) * y^(5j - 2) * z^(5k - 4) is in this sequence.

Examples

			6^2 = 2^2 + 2^5.
9^2 = 7^2 + 2^5.

Links

Crossrefs

Cf. A009003, A070067, A070745, A228946, A271576, A274035.

Programs

  • Mathematica
    c[n_]: = Count[n^2 - Range[(n^2 - 1)^(1/5)]^5, _?(IntegerQ[Sqrt[#]] &)] > 0;
Select[Range[1200], c]
  • PARI
    isok(n) = for (k=1, n-1, if (ispower(n^2-k^2, 5), return (1));); return (0); \\ Michel Marcus, Oct 06 2017

A293692 Numbers z such that x^2 + y^7 = z^2 for positive integers x and y.

Original entry on oeis.org

12, 18, 33, 54, 126, 160, 272, 366, 375, 520, 531, 540, 594, 630, 756, 825, 945, 1028, 1044, 1094, 1350, 1372, 1506, 1536, 1575, 1980, 2050, 2219, 2304, 2619, 2940, 3250, 3500, 3645, 3906, 3925, 4097, 4224, 4390, 4625, 5500, 5844, 5988, 6048, 6192, 6283, 6422
Offset: 1

Views

Author

XU Pingya, Oct 14 2017

Keywords

Comments

Let i, j and k are nonnegative integers, n is positive integer. As [(n^)^(7i+1) * (2n+1)^(7j + 3) * (n + 1)^(7k)]^2 + [n)^(2i) * (2n + 1)^(2j + 1) * (n + 1)^(2k)]^7 = [n^(7i) * (2n + 1)^(7j + 3) * (n+1)^(7k+1)]^2, so that number of form n^(7i) * (2n + 1)^(7j + 3) * (n+1)^(7k+1) is a term in sequence.
When (x, y, z) is solution of x^2 + y^3 = z^2 (i.e., z = A070745(n)), (x^(7i+1) * y^(7j + 2) * z^(7k)]^2, x^(2i) * y^(2j + 1) * z^(2k), x^(7i) * y^(7j + 2) * z^(7k+1) is solution of x^2 + y^7 = z^2.
When (x, y, z) is solution of x^2 + y^5 = z^2, (i.e., z = A293284(n)), x^(7i+1) * y^(7j + 1) * z^(7k), x^(2i) * y^(2j + 1) * z^(2k), x^(7i) * y^(7j + 1) * z^(7k+1) is solution of x^2 + y^7 = z^2.
When (x, y, z) is solution of x^2 + y^7 = z^2, (x^(7i+1) * y^(7j + 2) * z^(7k), x^(7i) * y^(j + 1) * z^(7k), x^(7i) * y^(7j +2) * z^(7k)) is also.
If x^2 + y^7 = z^2 then y^7 = z^2 - x^2 = (z - x)(z + x) and so (z - x, z + x) is a divisor pair of z^7. - David A. Corneth, May 24 2025

Examples

			4^2 + 2^7 = 12^2, 12 is a term.
31^2 + 2^7 = 33^2, 33 is a term.

Crossrefs

Cf. A070745, A228946, A293284.

Programs

  • Mathematica
    z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/7)]^7, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[6550], z]

A293690 Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.

Original entry on oeis.org

10, 17, 45, 80, 123, 136, 225, 234, 260, 270, 291, 325, 360, 365, 459, 510, 514, 640, 666, 745, 984, 1025, 1088, 1215, 1225, 1250, 1305, 1450, 1466, 1565, 1740, 1753, 1800, 1872, 1950, 1970, 2022, 2080, 2125, 2160, 2328, 2600, 2628, 2880, 2920, 3172, 3185
Offset: 1

Views

Author

XU Pingya, Oct 14 2017

Keywords

Comments

Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(3*i+1) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^6 = ((m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1))^2, so that the number of the form (m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1) is a term.
When (x, y, z) is a solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(3*i+1) * y^(3*j+1) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j+1) * z^(3*k+1)) is a solution of x^2 + y^6 = z^2.
When (x, y, z) is a solution of x^2 + y^6 = z^2, (x^(3*i+1) * y^(3*j) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j) * z^(3*k+1)) is also a solution of x^2 + y^6 = z^2.

Examples

			6^2 + 2^6 = 10^2, 10 is a term.
15^2 + 2^6 = 17^2, 17 is a term.

Crossrefs

Cf. A009003, A070745, A228946, A271576, A274035, A293692, A293694.

Programs

  • Mathematica
    z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/6)]^6, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[3200], z]

A372644 For n >= 1, a(n) is the least k >= 1 such that for some x >= 1, n^2 - k^2 = x^3 or a(n) = -1 if no such k exists.

Original entry on oeis.org

-1, -1, 1, -1, -1, 3, -1, -1, -1, 6, -1, -1, -1, 13, 3, -1, 15, -1, -1, -1, 15, -1, -1, 8, -1, -1, -1, 21, 25, -1, -1, -1, -1, -1, 15, 28, -1, -1, -1, -1, -1, 6, 11, -1, 36, -1, -1, 24, -1, -1, -1, -1, -1, -1, 45, -1, 39, -1, -1, 15, -1, 46, 35, -1, -1, 55
Offset: 1

Views

Author

Ctibor O. Zizka, May 08 2024

Keywords

Comments

For a given n this is a minimal solution of the Diophantine equation n^2 - k^2 = x^3 in positive integers. The solution exists only for n from A070745.

Examples

			n = 3: 9 - k^2 = x^3 is true for k = 1 and x = 2, thus a(3) = 1.
n = 6: 36 - k^2 = x^3 is true for k = 3 and x = 3, thus a(6) = 3.

Crossrefs

Cf. A070745.

Programs

  • Python
    from sympy import integer_nthroot
def A372644(n): return next((k for k in range(1,n) if integer_nthroot(n**2-k**2,3)[1]),-1) # Chai Wah Wu, May 11 2024
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.