A317746 Irregular triangle read by rows in which row n lists the divisors k of n such that k^n + n^k == 0 (mod k + n).

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 3, 6, 1, 7, 1, 2, 8, 1, 3, 9, 1, 10, 1, 11, 1, 4, 6, 12, 1, 13, 1, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 6, 9, 18, 1, 19, 1, 5, 20, 1, 3, 7, 21, 1, 22, 1, 23, 1, 3, 8, 12, 24, 1, 5, 25, 1, 13, 26, 1, 3, 9, 27, 1, 4, 28, 1, 29, 1, 6, 10, 15, 30

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 06 2018

Keywords

• nonn
• tabf

Comments

Triangle read by rows in which row n lists the type-1 divisors of n. For each divisor k of n, call k a type-r divisor of n if (r*k)^n + n^(r*k) == 0 (mod r*k + n), r >= 1.

Triangle read by rows in which row n lists the smallest types r of divisor k of n such that (r*k)^n + n^(r*k) == 0 (mod r*k + n) begins:

1;

1, 1;

1, 1;

1, 2, 1;

1, 1;

1, 3, 1, 1;

1, 1;

1, 1, 2, 1;

1, 1, 1;

1, 3, 2, 1;

1, 1;

1, 2, 3, 1, 1, 1;

..., where the total number of type-1 divisors of n is the sum of the number of all trivial divisors of n and a certain number of nontrivial divisors of n, namely: 1+0, 2+0, 2+0, 2+0, 2+0, 2+1, 2+0, 2+1, 2+1, 2+0, 2+0, 2+2, ...

Examples

			Triangle begins:
  1;
  1,  2;
  1,  3;
  1,  4;
  1,  5;
  1,  3,  6;
  1,  7;
  1,  2,  8;
  1,  3,  9;
  1, 10;
  1, 11;
  1,  4,  6, 12;
		

Crossrefs

Cf. A027750, A070824.

Programs

Magma

[[k: k in [ 1..n] | Denominator(n/k) eq 1 and Denominator((k^n+n^k)/(k+n)) eq 1]: n in [1..30]]

Mathematica

a[n_] := Select[ Divisors@ n, Mod[PowerMod[#, n, # + n] + PowerMod[n, #, # + n], # + n] == 0 &]; Array[a, 30] // Flatten (* Robert G. Wilson v, Aug 06 2018 *)