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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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%I #15 Feb 10 2023 12:54:39

%S 1,2,3,5,13,17,20,25,41,48,53,61,85,95,102,113,145,158,167,181,221,

%T 237,248,265,313,332,345,365,421,443,458,481,545,570,587,613,685,713,

%U 732,761,841,872,893,925

%N 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'.

%C 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.

%e 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.

%t mval[n_] := Floor[Sqrt[2 n] + 1/2]; (* A002024 *)

%t rval[n_] := (2 - 2 n + Round[Sqrt[2 n]] + Round[Sqrt[2 n]]^2)/2; (* A004736 *)

%t test[n_] := Mod[n - rval[n], mval[n]] == 0;

%t Select[Range[10000], test[#] &]

%Y Cf. A007607, A080856, A102083, A360416, A360417.

%K nonn,easy

%O 1,2

%A _James Propp_, Feb 06 2023