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.

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A360418 Numbers k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class a' mod b' (with r' in {1,...,m'}) iff m < m' or r > r'.

Original entry on oeis.org

1, 2, 3, 5, 13, 17, 20, 25, 41, 48, 53, 61, 85, 95, 102, 113, 145, 158, 167, 181, 221, 237, 248, 265, 313, 332, 345, 365, 421, 443, 458, 481, 545, 570, 587, 613, 685, 713, 732, 761, 841, 872, 893, 925, 1013, 1047, 1070, 1105, 1201, 1238, 1263, 1301, 1405, 1445, 1472, 1513, 1625, 1668, 1697, 1741, 1861
Offset: 1

Views

Author

James Propp, Feb 06 2023

Keywords

Comments

The sequence appears to be the interleaving of the four sequences A080856, A102083, A360416, A360417. This has been verified for values of k up to one million as of February 06 2023.
Above conjecture confirmed with more terms and linear recurrence. See supporting formula below. - Ray Chandler, Feb 10 2025

Examples

			The 1st congruence class in the list (with m=1 and r=1) is {1,2,3,...} which contains 1, so 1 is in the sequence. The 2nd congruence class (with m=2 and r=2) is {2,4,6,...} which contains 2, so 2 is in the sequence. The 3rd congruence class (with m=2 and r=1) is {1,3,5,...} which contains 3, so 3 is in the sequence. The 4th congruence class (with m=3 and r=3) is {3,6,9,...} which does not contain 4, so 4 is not in the sequence.

Links

Crossrefs

Cf. A002024, A004736, A007607, A080856, A102083, A360416, A360417.

Programs

  • Mathematica
    mval[n_] := Floor[Sqrt[2 n] + 1/2]; (* A002024 *)
rval[n_] := (2 - 2 n + Round[Sqrt[2 n]] + Round[Sqrt[2 n]]^2)/2; (* A004736 *)
test[n_] := Mod[n - rval[n], mval[n]] == 0;
Select[Range[10000], test[#] &]

Formula

From Ray Chandler, Feb 10 2025: (Start)
a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 8.
A080856(n) = A360417(n-1) + 2*A080856(n-1) - 2*A360417(n-2) - A080856(n-2) + A360417(n-3).
A102083(n) = A080856(n) + 2*A102083(n-1) - 2*A080856(n-1) - A102083(n-2) + A080856(n-2).
A360416(n) = A102083(n) + 2*A360416(n-1) - 2*A102083(n-1) - A360416(n-2) + A102083(n-2).
A360417(n) = A360416(n) + 2*A360417(n-1) - 2*A360416(n-1) - A360417(n-2) + A360416(n-2). (End)

A360416 a(n) = 8*n^2 - 9*n + 3.

Original entry on oeis.org

3, 2, 17, 48, 95, 158, 237, 332, 443, 570, 713, 872, 1047, 1238, 1445, 1668, 1907, 2162, 2433, 2720, 3023, 3342, 3677, 4028, 4395, 4778, 5177, 5592, 6023, 6470, 6933, 7412, 7907, 8418, 8945, 9488, 10047, 10622, 11213, 11820, 12443, 13082, 13737, 14408, 15095
Offset: 0

Views

Author

James Propp, Feb 06 2023

Keywords

Comments

This is one of the four quadratic sequences that, interleaved, yield A360418.

Links

Crossrefs

Cf. A360417, A360418.

Programs

  • Mathematica
    Table[8*n^2 - 9*n + 3, {n, 0, 100}]

Formula

From Elmo R. Oliveira, Jan 28 2025: (Start)
G.f.: (3 - 7*x + 20*x^2)/(1 - x)^3.
E.g.f.: (3 - x + 8*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = A360418(4*n-2) for n >= 1. (End)

Extensions

a(0)=3 prepended and more terms from Elmo R. Oliveira, Jan 28 2025
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