|
%I #34 Sep 08 2022 08:46:22
%S 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,
%T 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,
%U 12,24,1,5,25,1,13,26,1,3,9,27,1,4,28,1,29,1,6,10,15,30
%N 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).
%C 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.
%C 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:
%C 1;
%C 1, 1;
%C 1, 1;
%C 1, 2, 1;
%C 1, 1;
%C 1, 3, 1, 1;
%C 1, 1;
%C 1, 1, 2, 1;
%C 1, 1, 1;
%C 1, 3, 2, 1;
%C 1, 1;
%C 1, 2, 3, 1, 1, 1;
%C ..., 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, ...
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 3;
%e 1, 4;
%e 1, 5;
%e 1, 3, 6;
%e 1, 7;
%e 1, 2, 8;
%e 1, 3, 9;
%e 1, 10;
%e 1, 11;
%e 1, 4, 6, 12;
%t 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 *)
%o (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]]
%Y Cf. A027750, A070824.
%K nonn,tabf
%O 1,3
%A _Juri-Stepan Gerasimov_, Aug 06 2018
|
