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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.

%I A275280 #47 Mar 09 2017 05:27:31
%S A275280 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,
%T A275280 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,
%U A275280 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
%N 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.
%C A275280 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.
%C A275280 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.
%C A275280 The matrix a(n) is bounded by n, thus all products m <= n.
%C A275280 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.
%C A275280 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).
%C A275280 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.
%C A275280 a(p^e) = A027750(p^e) = A162306(p^e) = A275055(p^e) for e >= 1.
%H A275280 Michael De Vlieger, <a href="/A275280/b275280.txt">Table of n, a(n) for n = 1..17689</a> (Rows 1 <= n <= 1000)
%e A275280 Triangle begins:
%e A275280 1;
%e A275280 1, 2;
%e A275280 1, 3;
%e A275280 1, 2, 4;
%e A275280 1, 5;
%e A275280 1, 2, 4, 3, 6;
%e A275280 1, 7;
%e A275280 1, 2, 4, 8;
%e A275280 1, 3, 9;
%e A275280 1, 2, 4, 8, 5, 10;
%e A275280 1, 11;
%e A275280 1, 2, 4, 8, 3, 6, 12, 9;
%e A275280 1, 13;
%e A275280 1, 2, 4, 8, 7, 14;
%e A275280 1, 3, 9, 5, 15;
%e A275280 1  2, 4, 8, 16;
%e A275280 1, 17;
%e A275280 1, 2, 4, 8, 16, 3, 6, 12, 9, 18;
%e A275280 ...
%e A275280 2 prime divisors: n = 96
%e A275280    1  2  4  8 16 32 64
%e A275280    3  6 12 24 48 96
%e A275280    9 18 36 72
%e A275280   27 54
%e A275280   81
%e A275280 thus a(96) = {1,2,4,8,16,32,64,3,6,12,24,48,96,9,18,36,72,27,54,81}.
%e A275280 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.
%e A275280 3 prime divisors: n = 60
%e A275280 (the 3 dimensional levels correspond with powers of 5)
%e A275280    level 5^0:            level 5^1:         level 5^2:
%e A275280    1  2  4  8 16 32  |    5 10 20 40    |   25 50
%e A275280    3  6 12 24 48     |   15 30 60       |
%e A275280    9 18 36           |   45             |
%e A275280   27 54              |                  |
%e A275280 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}.
%e A275280 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.
%t A275280 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 *)
%Y A275280 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).
%K A275280 nonn,easy,tabf
%O A275280 1,3
%A A275280 _Michael De Vlieger_, Jul 28 2016