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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.

User: Edward Krogius

Edward Krogius's wiki page.

Edward Krogius has authored 3 sequences.

A356251 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+1)/12.

Original entry on oeis.org

0, 6, 50, 210, 630, 1540, 3276, 6300, 11220, 18810, 30030, 46046, 68250, 98280, 138040, 189720, 255816, 339150, 442890, 570570, 726110, 913836, 1138500, 1405300, 1719900, 2088450, 2517606, 3014550, 3587010, 4243280, 4992240, 5843376, 6806800, 7893270, 9114210
Offset: 0

Views

Author

Edward Krogius, Jul 31 2022

Keywords

Comments

Sum of all numbers squared in ordered triples (x,y,z) where 0 <= x <= y <= z <= n.

Examples

			a(1) = 6 because we have the triples (0,0,0), (0,0,1), (0,1,1), (1,1,1).

Links

Crossrefs

Cf. A033487.

Programs

  • Mathematica
    Table[n*(n + 1)*(n + 2)*(n + 3)*(2*n + 1)/12, {n, 0, 35}] (* Amiram Eldar, Sep 11 2022 *)
Table[Sum[x^2 + y^2 + z^2, {x, 0, g}, {y, x, g}, {z, y, g}], {g, 0, 30}] (* Horst H. Manninger, Jun 19 2025 *)

Formula

G.f.: 2*x*(7*x+3)/(x-1)^6.
From Amiram Eldar, Sep 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 136/15 - 64*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*Pi/5 - 32*log(2)/5 - 82/15. (End)

A339026 Number of pairs (x,y): 1 <= x < y <= n*n, such that x*y is a square.

Original entry on oeis.org

0, 1, 4, 8, 16, 27, 40, 58, 80, 105, 128, 158, 198, 237, 278, 336, 382, 435, 502, 574, 632, 699, 798, 868, 968, 1063, 1176, 1262, 1362, 1505, 1590, 1732, 1880, 2007, 2150, 2272, 2434, 2567, 2746, 2930, 3062, 3265, 3444, 3654, 3832, 4035, 4284, 4454, 4668, 4875, 5142, 5364, 5590, 5889, 6078, 6336, 6614, 6881, 7166
Offset: 1

Views

Author

Edward Krogius, Nov 19 2020

Keywords

Examples

			For n = 3, we have the following solutions: (1,4), (1,9), (2,8), (4,9), therefore a(3) = 4.
For n = 4, we have the following solutions: (1,4), (1,9), (1,16), (2,8), (3,12), (4,9), (4,16), (9,16), therefore a(4) = 8.

Links

Crossrefs

Cf. A000010, A015614, A018805, A132188, A132189, A132345, A338894.

Programs

  • Mathematica
    Array[Sum[EulerPhi[j] Floor[(#^2)/(j^2)], {j, 2, #}] &, 59] (* Michael De Vlieger, Dec 11 2020 *)
  • PARI
    A339026(n) = sum(i=2,n,floor(n^2/i^2)*eulerphi(i)); \\ Antti Karttunen, Nov 23 2020

Formula

a(n) = Sum_{j=2..n} phi(j) * floor(n^2/j^2).
a(n) = (A338894(n) - n^2)/2.
a(n) = A132189(n^2)/2. - Antti Karttunen, Nov 23 2020

A338894 Number of ordered pairs (x,y): 1 <= x, y <= n*n, such that x*y is a square.

Original entry on oeis.org

1, 6, 17, 32, 57, 90, 129, 180, 241, 310, 377, 460, 565, 670, 781, 928, 1053, 1194, 1365, 1548, 1705, 1882, 2125, 2312, 2561, 2802, 3081, 3308, 3565, 3910, 4141, 4488, 4849, 5170, 5525, 5840, 6237, 6578, 7013, 7460
Offset: 1

Views

Author

Edward Krogius, Nov 14 2020

Keywords

References

  • The Finnish National Upper secondary Matriculation Examination Long Maths Problem #12 (Mar 18th, 2020) included finding all gridpoints in a [1..100]x[1..100] grid with an integer geometric mean sparked some national interest in gcd integer sequences and their generating algorithms.

Links

Crossrefs

Cf. A000010, A057918, A132188, A339026.

Programs

  • PARI
    A338894(n) = sum(i=1,n*n,sum(j=1,n*n,issquare(i*j))); \\ (Naive implementation) - Antti Karttunen, Nov 23 2020

Formula

a(n) = 2*A339026(n) + n^2.
a(n) = A132188(n^2). - Antti Karttunen, Nov 23 2020

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