A360416 -id:A360416 - cp's OEIS frontend

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'.

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

James Propp, Feb 06 2023

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

			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.

Ray Chandler, Table of n, a(n) for n = 1..14142 (terms up to 10^8)
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 2, -2, 0, 0, -1, 1).

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

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[#] &]

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)

A360417 a(n) = 8n^2 - 7n + 2.

Original entry on oeis.org

2, 3, 20, 53, 102, 167, 248, 345, 458, 587, 732, 893, 1070, 1263, 1472, 1697, 1938, 2195, 2468, 2757, 3062, 3383, 3720, 4073, 4442, 4827, 5228, 5645, 6078, 6527, 6992, 7473, 7970, 8483, 9012, 9557, 10118, 10695, 11288, 11897, 12522, 13163, 13820, 14493, 15182

Offset: 0

James Propp, Feb 06 2023

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

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051870, A125201, A360416, A360418.

Mathematica
```
Table[8*n^2 - 7*n + 2, {n, 0, 100}]
```

From Elmo R. Oliveira, Jan 28 2025: (Start)
G.f.: (17*x^2-3*x+2)/(1-x)^3.
E.g.f.: exp(x)*(2 + x + 8*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = A360418(4*n-1) for n>=1. (End)
a(n) = A125201(n) + 1, n >= 1. - Omar E. Pol, Jan 28 2025

a(0)=2 prepended and more terms from Alois P. Heinz, Jan 28 2025

cp's OEIS Frontend

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'.

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A360417 a(n) = 8n^2 - 7n + 2.

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cp's OEIS Frontend

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'.

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Author

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Examples

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Crossrefs

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Mathematica

Formula

A360417 a(n) = 8*n^2 - 7*n + 2.

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Author

Keywords

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Crossrefs

Programs

Mathematica

Formula

Extensions

A360417 a(n) = 8n^2 - 7n + 2.