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

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, 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, 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

Offset: 1

Author

Juri-Stepan Gerasimov, Mar 25 2017

Keywords

• nonn
• tabf

Comments

Row n lists divisors of n that are divisible by A007947(n). - Robert Israel, Apr 27 2017

Examples

			Triangle begins:
    1;
    2;
    3;
    2, 4;
    5;
    6;
    7;
    2, 4, 8;
    3, 9;
    10;
    11;
    6, 12;
    13;
    14;
    15;
    2, 4, 8, 16.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10002 (rows 1 to 5250, flattened)

Crossrefs

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).

Cf. A003557, A027750.

Programs

Magma

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

Maple

f:= proc(n) local r;
    r:= convert(numtheory:-factorset(n),`*`);
    op(sort(convert(map(`*`, numtheory:-divisors(n/r),r),list)))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 27 2017

Mathematica

Flatten[Table[Select[Range[n], Divisible[n, #] && Divisible[#^n, n] &], {n, 50}]] (* Indranil Ghosh, Mar 25 2017 *)

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

Python

for n in range(1, 51):
    print([i for i in range(1, n + 1) if n%i==0 and (i**n)%n==0]) # Indranil Ghosh, Mar 25 2017

Formula

T(n,k) = A007947(n) * A027750(A003557(n), k). - Robert Israel, Apr 27 2017