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A275055 Irregular triangle read by rows listing divisors d of n in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes. 4
1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 4, 3, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 4, 8, 3, 6, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(p^e) = A027750(p^e) for e >= 1.

The matrix of products that are divisors of n is arranged such that the powers of the prime divisors range across an axis, one axis per prime divisor. Thus a squarefree semiprime has a 2-dimensional matrix, a sphenic number has 3 dimensions, etc.

Generally, the number of dimensions for the matrix of divisors = omega(n) = A001221(n). Because of this, tau(n)*(mod omega(n)) = 0 for n > 1.

This follows from the formula for tau(n).

Prime divisors p of n are considered in numerical order.

Product matrix of tensors T = 1,p,p^2,...,p^e that include the powers 1 <= e of the prime divisor p that divide n.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..11214 (Rows 1 <= n <= 1500)

Eric Weisstein's World of Mathematics, Divisor

EXAMPLE

Triangle begins:

1;

1, 2;

1, 3;

1, 2, 4;

1, 5;

1, 2, 3, 6;

1, 7;

1, 2, 4, 8;

1, 3, 9;

1, 2, 5, 10;

1, 11;

1, 2, 4, 3, 6, 12;

1, 13;

1, 2, 7, 14;

1, 3, 5, 15;

1  2, 4, 8, 16;

1, 17;

1, 2, 3, 6, 9, 18;

...

2 prime divisors: n = 72

   1  2  4  8

   3  6 12 24

   9 18 36 72

   thus a(72) = {1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72}

3 prime divisors: n = 60

(the 3 dimensional levels correspond with powers of 5)

  level 5^0:        level 5^1:

   1  2  4    |     5  10  20

   3  6 12    |    15  30  60

   thus a(60) = {1, 2, 4, 3, 6, 12, 5, 10, 20, 15, 30, 60}

4 prime divisors: n = 210

(the 3 dimensional levels correspond with powers of 5,

the 4 dimensional levels correspond with powers of 7)

  level 5^0*7^0:    level 5^1*7^0:

     1   2     |     5  10

     3   6     |    15  30

  level 5^0*7^1:    level 5^1*7^1:

     7  14     |    35  70

    21  42     |   105 210

   thus a(210) = {1,2,3,6,5,10,15,30,7,14,21,42,35,70,105,210}

MATHEMATICA

{{1}}~Join~Table[TensorProduct @@ Reverse@ Apply[PowerRange[1, #1^#2, #1] &, # &@ FactorInteger@ n, 1], {n, 2, 30}] // Flatten

CROSSREFS

Cf. A027750, A000005 (row length), A000203 (row sums), A056538.

Sequence in context: A162306 A233773 A027750 * A254679 A275280 A087295

Adjacent sequences:  A275052 A275053 A275054 * A275056 A275057 A275058

KEYWORD

nonn,easy,tabf

AUTHOR

Michael De Vlieger, Jul 14 2016

STATUS

approved

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Last modified October 21 11:39 EDT 2018. Contains 316414 sequences. (Running on oeis4.)