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A329534 Irregular triangle read by rows: for n >= 1 row n lists the k from [1, 2, ... , n] such that A002378(k-1) = (k-1)*k == 0 (mod n).

%I A329534 #56 Sep 08 2022 08:46:24
%S A329534 1,1,2,1,3,1,4,1,5,1,3,4,6,1,7,1,8,1,9,1,5,6,10,1,11,1,4,9,12,1,13,1,
%T A329534 7,8,14,1,6,10,15,1,16,1,17,1,9,10,18,1,19,1,5,16,20,1,7,15,21,1,11,
%U A329534 12,22,1,23,1,9,16,24,1,25
%N A329534 Irregular triangle read by rows: for n >= 1 row n lists the k from [1, 2, ... , n] such that A002378(k-1) = (k-1)*k == 0 (mod n).
%C A329534 n-th row length gives 1 for n = 1, and 2^A001221(n) for n >= 2 , that is A034444(n). [Proof: Unique lifting theorem (e.g., Apostol, 5.30 (a), p.121) for this congruence, and only two solutions 1 and p for primes p. See also the Yuval Dekel, Sep 21 2003, comment in  A034444. - _Wolfdieter Lang_, Feb 05 2020]
%D A329534 Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
%e A329534 The irregular triangle T(n,k) begins
%e A329534 n\k  1  2  3  4 ...
%e A329534 1:   1
%e A329534 2:   1  2
%e A329534 3:   1  3
%e A329534 4:   1  4
%e A329534 5:   1  5
%e A329534 6:   1  3  4  6
%e A329534 7:   1  7
%e A329534 8:   1  8
%e A329534 9:   1  9
%e A329534 10:  1  5  6 10
%e A329534 11:  1 11
%e A329534 12:  1  4  9 12
%e A329534 13:  1 13
%e A329534 14:  1  7  8 14
%e A329534 15:  1  6 10 15
%e A329534 16:  1 16
%e A329534 17:  1 17
%e A329534 18:  1  9 10 18
%e A329534 19:  1 19
%e A329534 20:  1  5 16 20
%e A329534 ...
%t A329534 Table[Select[Range@ n, Mod[-n + # (# - 1), n] == 0 &], {n, 25}] // Flatten (* _Michael De Vlieger_, Nov 18 2019 *)
%o A329534 (Magma) [[k: k in [1..n] | k^2 mod n eq k]: n in [1..38]];
%o A329534 (PARI) row(n) = select(x->(Mod(x, n) == Mod(x, n)^2), [1..n]); \\ _Michel Marcus_, Nov 19 2019
%Y A329534 Cf. A000010, A000225, A000688, A000961, A001221, A006881, A006530, A007875, A020639, A024619, A034444, A077610, A135972, A309307.
%K A329534 nonn,easy,tabf
%O A329534 1,3
%A A329534 _Juri-Stepan Gerasimov_, Nov 15 2019
%E A329534 Edited by _Wolfdieter Lang_, Feb 05 2020