

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 2dimensional 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



