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Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).
3

%I #7 Sep 27 2024 05:42:37

%S 3,6,7,14,28,31,62,124,248,496,127,254,508,1016,2032,4064,8128,8191,

%T 16382,32764,65528,131056,262112,524224,1048448,2096896,4193792,

%U 8387584,16775168,33550336,131071,262142,524284,1048568,2097136,4193792

%N Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

%C Also, row n list the divisors of n-th perfect number that are not powers of 2.

%C First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.

%e Triangle begins:

%e 3, 6,

%e 7, 14, 28

%e 31, 62, 124, 248, 496

%e 127, 254, 508, 1016, 2032, 4064, 8128

%e ...

%e ==========================================================

%e Row .... First term ..... Last term ....... Row sum ......

%e n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .

%e ==========================================================

%e 1 ............ 3 .............. 6 ......... 3^2 = 9

%e 2 ............ 7 ............. 28 ......... 7^2 = 49

%e 3 ........... 31 ............ 496 ........ 31^2 = 961

%e 4 .......... 127 ........... 8128 ....... 127^2 = 16129

%e 5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481

%Y Cf. A000043, A000396, A000668, A018254, A018487, A133024, A133025, A133031, A133049, A135652, A135653, A135654, A135655.

%K nonn,tabf

%O 1,1

%A _Omar E. Pol_, Apr 22 2008