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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.

%I A275055 #13 Feb 16 2025 08:33:36
%S A275055 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,
%T A275055 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,
%U A275055 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
%N 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.
%C A275055 a(p^e) = A027750(p^e) for e >= 1.
%C A275055 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.
%C A275055 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.
%C A275055 This follows from the formula for tau(n).
%C A275055 Prime divisors p of n are considered in numerical order.
%C A275055 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.
%H A275055 Michael De Vlieger, <a href="/A275055/b275055.txt">Table of n, a(n) for n = 1..11214</a> (Rows 1 <= n <= 1500)
%H A275055 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>
%e A275055 Triangle begins:
%e A275055 1;
%e A275055 1, 2;
%e A275055 1, 3;
%e A275055 1, 2, 4;
%e A275055 1, 5;
%e A275055 1, 2, 3, 6;
%e A275055 1, 7;
%e A275055 1, 2, 4, 8;
%e A275055 1, 3, 9;
%e A275055 1, 2, 5, 10;
%e A275055 1, 11;
%e A275055 1, 2, 4, 3, 6, 12;
%e A275055 1, 13;
%e A275055 1, 2, 7, 14;
%e A275055 1, 3, 5, 15;
%e A275055 1  2, 4, 8, 16;
%e A275055 1, 17;
%e A275055 1, 2, 3, 6, 9, 18;
%e A275055 ...
%e A275055 2 prime divisors: n = 72
%e A275055    1  2  4  8
%e A275055    3  6 12 24
%e A275055    9 18 36 72
%e A275055    thus a(72) = {1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72}
%e A275055 3 prime divisors: n = 60
%e A275055 (the 3 dimensional levels correspond with powers of 5)
%e A275055   level 5^0:        level 5^1:
%e A275055    1  2  4    |     5  10  20
%e A275055    3  6 12    |    15  30  60
%e A275055    thus a(60) = {1, 2, 4, 3, 6, 12, 5, 10, 20, 15, 30, 60}
%e A275055 4 prime divisors: n = 210
%e A275055 (the 3 dimensional levels correspond with powers of 5,
%e A275055 the 4 dimensional levels correspond with powers of 7)
%e A275055   level 5^0*7^0:    level 5^1*7^0:
%e A275055      1   2     |     5  10
%e A275055      3   6     |    15  30
%e A275055   level 5^0*7^1:    level 5^1*7^1:
%e A275055      7  14     |    35  70
%e A275055     21  42     |   105 210
%e A275055    thus a(210) = {1,2,3,6,5,10,15,30,7,14,21,42,35,70,105,210}
%t A275055 {{1}}~Join~Table[TensorProduct @@ Reverse@ Apply[PowerRange[1, #1^#2, #1] &, # &@ FactorInteger@ n, 1], {n, 2, 30}] // Flatten
%Y A275055 Cf. A027750, A000005 (row length), A000203 (row sums), A056538.
%K A275055 nonn,easy,tabf
%O A275055 1,3
%A A275055 _Michael De Vlieger_, Jul 14 2016