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

OEIS Server: https://oeis.org/edit/global/2944

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Table[Select[Range@ n, Mod[-n + #^2, + # (# - 1), n] == # &], {] == 0 &], {n, 3925}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

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Michael De Vlieger: Fixed Mathematica per Wolfdieter's pink box comment immediately preceding this one.

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

1, 1, 2, 1, 3, 1, 4, 2, 31, 5, 6, 71, 83, 4, 56, 91, 107, 31, 8, 11, 12, 61, 79, 131, 5, 6, 910, 141, 1511, 161, 84, 9, 1712, 181, 413, 151, 197, 68, 14, 20, 10, 11, 211, 226, 810, 15, 23, 241, 1216, 131, 2517, 261, 79, 2010, 2718, 281, 519, 91, 145, 1516, 20, 24, 29, 301, 317, 1115, 21, 32, 16, 17, 33, 14, 201, 3411, 812, 2722, 351, 3623, 181, 199, 3716, 1224, 261, 3825

1,23

n-th row length gives 1 for A309307(n) = A000225( = 1, and 2^A001221(n) for n)) = >= 2 , that is A034444(n) - 1). [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]

A000961(n)-th row length is a nonprime number of the form 2^x - 1.

A024619(n)-th row length is a Mersenne prime.

A006881(n)-th row length = 3.

n-th row sum gives (n-1)*A007875(n) = (n-1)*A000010(2^A001221(n)).

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

n\k 1 2 3 ... 4 ...

1: 1:

2: : 1 2

3: 2

4: 3

5: 4

6: 2 3 5

7: 6

8: 7

9: 8

10: 4 5 9

3: 1 3

4: 1 4

5: 1 5

6: 1 3 4 6

7: 1 7

8: 1 8

119: 1 9

12: 3 8 11

13: 12

14: 6 7 13

15: 5 9 14.

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

...

Table[Select[Range@ n, Mod[[-n - #^ + #^2, n] == # &], {n, 39}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

(MAGMA) [[k: k in [1..n-1] | -] | k^2 mod n eq k]: n in [1..38]];

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

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

nonn,easy,tabf

Edited by Wolfdieter Lang, Feb 05 2020

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Wolfdieter Lang: Since there came no answer from Juri-Stepan Gerasimov I edited this sequence by changing k -> k-1, and the range of k is now [1...n] (not [1..n-1]. I used [1..n] insted of [0..(n-1)] in order to compare this with the ireegular triangle A077610.// I changed the MAGMA nad PARI program accordingly ( is it ok?) , and wrote to Michael De Vlieger to change the Mma program.

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Sean A. Irvine: This doesn't look like a triangle to me, not even an irregular one.

Wolfdieter Lang: To Juri-Stepan Gerasimov: with some small redefinition this seems to become the irregular triangle  A077610 (n,  m) ( n-th row lists unitary divisors of n) with offset n >= 1, and row length A034444(n). If true I suggest to recycle this proposal.  / /
The redefinition is as follows:  Do not consider  A002378(k)  =   k^2 + k == 0 (mod n), with representative k (from {0, 1 ,...,n-1}, but use   A002378(k-1) = (k-1)*k == 0 (mod n), n >= 1 and  k from {1, 2, ... ,n}. Then this becomes the irregular triangle T(n, m)  =   A077610 (n,  m) , with m = 1, 2, ..., A034444(n). 
Do you agree?

Wolfdieter Lang: To Juri-Stepan Gerasimov: Please erase in my comment the 'seems to', and replace 'become' by 'becomes', because gcd(k, (k-1)*k/k) = gcd(k, k-1) = gcd( k, (k-1) -1*k) = gcd(k, -1)  = gcd(k, 1) = 1. Therefore, this entry should be recycled, and one could instead give a comment in  A077610 like "Row n gives the integers k with 1 <= k <= n and   A002378(k-1) = K^2 -k == 0 (mod n), for n >= 1."

Wolfdieter Lang: It should have been: A002378(k-1) = k^2 -k == 0 (mod n). Sorry.

Wolfdieter Lang: To  To Juri-Stepan Gerasimov: Please forget all my comments above. I am very, very sorry. I was thinking about a different irregular triangle, namely [[1], [1, 2], [1, 3], [1, 4], [1, 5], [1, 3, 4, 6], [1, 7], [1, 8], [1, 9], [1, 5, 6, 10] , ...] which you could consider instead of the one you gave. But this is NOT  A077610 (compare row n=6) . This seems not to be in OEIS.  The definition is:  Row n gives the integers k with 1 <= k <= n and A002378(k-1) = k^2 - k == 0 (mod n), for n >= 1. The row length seems to be  A034444(n).  What do you think about giving this triangle instead of the one you gave?

Wolfdieter Lang: To the editors: I have writen an e-mail to Juri-Stepan Gerasimov (Jan 10).

Wolfdieter Lang: Today I have repeated the mail .

A024619(n)-th row length is a Mersenne prime.

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A000961(n)-th row length <= is a nonprime number of the form 2^x - 1.

A024619(n)-th row length is Mersenne prime.

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

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