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.

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

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: A368194 A233773 A027750 * A254679 A343651 A355634

Adjacent sequences: A275052 A275053 A275054 * A275056 A275057 A275058

Keyword

nonn,easy,tabf

Author

Michael De Vlieger, Jul 14 2016

Status

approved