A275280 Irregular triangle listing numbers m of n that have prime divisors p that also divide 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, 4, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 8, 5, 10, 1, 11, 1, 2, 4, 8, 3, 6, 12, 9, 1, 13, 1, 2, 4, 8, 7, 14, 1, 3, 9, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 4, 8, 16, 3, 6, 12, 9, 18, 1, 19, 1, 2, 4, 8, 16, 5, 10, 20, 1, 3, 9, 7, 21, 1, 2, 4, 8, 16, 11, 22, 1, 23

Offset: 1

Author

Michael De Vlieger, Jul 28 2016

Keywords

• nonn
• easy
• tabf

Comments

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

This sequence is analogous to A275055 but differs from it in that the tensors T include not only powers p^e that divide n but all powers p^e <= n.

The matrix a(n) is bounded by n, thus all products m <= n.

Let omega(n) = A001221(n). The matrix a(n) has omega(n) dimensions and is an omega(n)-dimensional simplex with (omega(n)-1) "right-angle: sides and 1 irregular surface that is bounded by n.

A027750(n) is a subset of A162306(n) and in a(n), the terms of A275055(n) appear in an contiguous omega(n)-dimensional parallelepiped (parallelotope) with 1 at the origin and n at the opposite corner. Thus the omega(n)-dimensional array described by A275055(n) is fully contained in the simplex-like matrix described by a(n). Divisors appear within the parallelepiped while nondivisors appear in the field outside the parallelepiped (see examples). Terms within the parallelepiped appear in A027750(n) while those outside appear in A272618(n).

For a(2^x + 2) there is a term m = (n-2); m != (n-1) except for n=2, since GCD(n, n-1)=1.

a(p^e) = A027750(p^e) = A162306(p^e) = A275055(p^e) for e >= 1.

Examples

			Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 5;
1, 2, 4, 3, 6;
1, 7;
1, 2, 4, 8;
1, 3, 9;
1, 2, 4, 8, 5, 10;
1, 11;
1, 2, 4, 8, 3, 6, 12, 9;
1, 13;
1, 2, 4, 8, 7, 14;
1, 3, 9, 5, 15;
1  2, 4, 8, 16;
1, 17;
1, 2, 4, 8, 16, 3, 6, 12, 9, 18;
...
2 prime divisors: n = 96
   1  2  4  8 16 32 64
   3  6 12 24 48 96
   9 18 36 72
  27 54
  81
thus a(96) = {1,2,4,8,16,32,64,3,6,12,24,48,96,9,18,36,72,27,54,81}.
The divisors of 72 (thus the terms of A275055(72)) appear in a rectangle delimited by 1 at top left and 72 at bottom right.
3 prime divisors: n = 60
(the 3 dimensional levels correspond with powers of 5)
   level 5^0:            level 5^1:         level 5^2:
   1  2  4  8 16 32  |    5 10 20 40    |   25 50
   3  6 12 24 48     |   15 30 60       |
   9 18 36           |   45             |
  27 54              |                  |
thus a(60) = {1,2,4,8,16,32,3,6,12,24,48,9,18,36,27,54,5,10,20,40,15,30,60,45,25,50}.
The divisors of 60 (thus the terms of A275055(60)) appear in a parallelepiped delimited by 1 at top left of level 5^0 and 60 at bottom right of level 5^1.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..17689 (Rows 1 <= n <= 1000)

Crossrefs

Cf. A162306, A010846 (row length), A243103 (row product), A027750 (divisors of n), A000005 (number of divisors of n), A272618 (nondivisors m <= n that have prime divisors p that also divide n), A243822 (number of such nondivisors of n), A275055 (Product of tensor of prime divisor powers that are also divisors).

Programs

Mathematica

f[n_] := If[n == 1, 1, Function[w, ToExpression@ StringJoin["With[{n=", ToString@ n, "}, Table[", ToString@ InputForm[Times @@ Map[Power @@ # &, w]], ", ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w]]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, Reverse[FactorInteger[n][[All, 1]]]] ]; Array[f, 24] // Flatten (* Michael De Vlieger, Mar 08 2017 *)