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A284318 Triangle read by rows in which row n lists divisors d of n such that n divides d^n.

%I A284318 #48 Feb 25 2025 10:58:05
%S A284318 1,2,3,2,4,5,6,7,2,4,8,3,9,10,11,6,12,13,14,15,2,4,8,16,17,6,18,19,10,
%T A284318 20,21,22,23,6,12,24,5,25,26,3,9,27,14,28,29,30,31,2,4,8,16,32,33,34,
%U A284318 35,6,12,18,36,37,38,39,10,20,40,41,42,43,22,44,15,45,46,47,6,12,24,48,7,49,10,50
%N A284318 Triangle read by rows in which row n lists divisors d of n such that n divides d^n.
%C A284318 Row n lists divisors of n that are divisible by A007947(n). - _Robert Israel_, Apr 27 2017
%H A284318 Robert Israel, <a href="/A284318/b284318.txt">Table of n, a(n) for n = 1..10002</a>  (rows 1 to 5250, flattened)
%F A284318 T(n,k) = A007947(n) * A027750(A003557(n), k). - _Robert Israel_, Apr 27 2017
%e A284318 Triangle begins:
%e A284318     1;
%e A284318     2;
%e A284318     3;
%e A284318     2, 4;
%e A284318     5;
%e A284318     6;
%e A284318     7;
%e A284318     2, 4, 8;
%e A284318     3, 9;
%e A284318     10;
%e A284318     11;
%e A284318     6, 12;
%e A284318     13;
%e A284318     14;
%e A284318     15;
%e A284318     2, 4, 8, 16.
%p A284318 f:= proc(n) local r;
%p A284318     r:= convert(numtheory:-factorset(n),`*`);
%p A284318     op(sort(convert(map(`*`, numtheory:-divisors(n/r),r),list)))
%p A284318 end proc:
%p A284318 map(f, [$1..100]); # _Robert Israel_, Apr 27 2017
%t A284318 Flatten[Table[Select[Range[n], Divisible[n, #] && Divisible[#^n, n] &], {n, 50}]] (* _Indranil Ghosh_, Mar 25 2017 *)
%o A284318 (Magma) [[u: u in [1..n] | Denominator(n/u) eq 1 and Denominator(u^n/n) eq 1]: n in [1..50]];
%o A284318 (PARI) for(n=1, 50, for(i=1, n, if(n%i==0 & (i^n)%n==0, print1(i,", "););); print();); \\ _Indranil Ghosh_, Mar 25 2017
%o A284318 (Python)
%o A284318 for n in range(1, 51):
%o A284318     print([i for i in range(1, n + 1) if n%i==0 and (i**n)%n==0]) # _Indranil Ghosh_, Mar 25 2017
%Y A284318 Cf. A000961 (1 together with k such that k divides p^k for some prime divisor p of k), A005361 (row length), A007774 (m such that m divides s^m for some semiprime divisor s of m), A007947 (smallest u such that u^n|n and n|u, or divisor k such that A000005(k) = 2^A001221(n)), A057723 (row sums), A066503 (difference between largest x and smallest y such that x^n|n, n|x, y^n|n and n|y).
%Y A284318 Cf. A003557, A027750.
%K A284318 nonn,tabf
%O A284318 1,2
%A A284318 _Juri-Stepan Gerasimov_, Mar 25 2017