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).

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, 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, 12, 22, 1, 23, 1, 9, 16, 24, 1, 25

Offset: 1

Author

Juri-Stepan Gerasimov, Nov 15 2019

Keywords

• nonn
• easy
• tabf

Comments

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]

Examples

			The irregular triangle T(n,k) begins
n\k  1  2  3  4 ...
1:   1
2:   1  2
3:   1  3
4:   1  4
5:   1  5
6:   1  3  4  6
7:   1  7
8:   1  8
9:   1  9
10:  1  5  6 10
11:  1 11
12:  1  4  9 12
13:  1 13
14:  1  7  8 14
15:  1  6 10 15
16:  1 16
17:  1 17
18:  1  9 10 18
19:  1 19
20:  1  5 16 20
...
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.

Crossrefs

Cf. A000010, A000225, A000688, A000961, A001221, A006881, A006530, A007875, A020639, A024619, A034444, A077610, A135972, A309307.

Programs

Magma

[[k: k in [1..n] | k^2 mod n eq k]: n in [1..38]];

Mathematica

Table[Select[Range@ n, Mod[-n + # (# - 1), n] == 0 &], {n, 25}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI

row(n) = select(x->(Mod(x, n) == Mod(x, n)^2), [1..n]); \\ Michel Marcus, Nov 19 2019

Extensions

Edited by Wolfdieter Lang, Feb 05 2020