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Triangle read by rows in which row n lists divisors d of n such that n divides d^n.
5

%I #44 Sep 08 2022 08:46:19

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

%C Row n lists divisors of n that are divisible by A007947(n). - _Robert Israel_, Apr 27 2017

%H Robert Israel, <a href="/A284318/b284318.txt">Table of n, a(n) for n = 1..10002</a> (rows 1 to 5250, flattened)

%F T(n,k) = A007947(n) * A027750(A003557(n), k). - _Robert Israel_, Apr 27 2017

%e Triangle begins:

%e 1;

%e 2;

%e 3;

%e 2, 4;

%e 5;

%e 6;

%e 7;

%e 2, 4, 8;

%e 3, 9;

%e 10;

%e 11;

%e 6, 12;

%e 13;

%e 14;

%e 15;

%e 2, 4, 8, 16.

%p f:= proc(n) local r;

%p r:= convert(numtheory:-factorset(n),`*`);

%p op(sort(convert(map(`*`, numtheory:-divisors(n/r),r),list)))

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Apr 27 2017

%t Flatten[Table[Select[Range[n], Divisible[n, #] && Divisible[#^n, n] &], {n, 50}]] (* _Indranil Ghosh_, Mar 25 2017 *)

%o (Magma) [[u: u in [1..n] | Denominator(n/u) eq 1 and Denominator(u^n/n) eq 1]: n in [1..50]];

%o (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 (Python)

%o for n in range(1, 51):

%o ....print [i for i in range(1, n + 1) if n%i==0 and (i**n)%n==0] # _Indranil Ghosh_, Mar 25 2017

%Y 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 Cf. A003557, A027750.

%K nonn,tabf

%O 1,2

%A _Juri-Stepan Gerasimov_, Mar 25 2017