A162306 Irregular triangle in which row n contains the numbers <= n whose prime factors are a subset of prime factors of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 8, 10, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 1, 13, 1, 2, 4, 7, 8, 14, 1, 3, 5, 9, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 1, 19, 1, 2, 4, 5, 8, 10, 16, 20, 1, 3, 7, 9, 21, 1, 2, 4, 8, 11, 16, 22, 1, 23

Offset: 1

Author

T. D. Noe, Jun 30 2009

Keywords

• nonn
• tabf
• look

Comments

Row n begins with 1, ends with n, and has A010846(n) terms.

From Michael De Vlieger, Jul 08 2014: (Start)

Prime p has {1, p} and A010846(p) = 2.

Prime power p^e has {1, p, ..., p^e} and A010846(p^e) = A000005(p^e) = e + 1.

Composite c that are not prime powers have A010846(c) = A000005(c) + A243822(c), where A243822(c) is nonzero positive, since the minimum prime divisor p of c produces at least one semidivisor (e.g., p^2 < c). Thus these have the set of divisors of c and at least one semidivisor p^2. For squareful c that are not prime powers, p^2 may divide c, but p^3 does not. The minimum squareful c = 12, 2^3 does not divide 12 yet is less than 12 and is a product of the minimum prime divisor of 12. All other even squareful c admit a power of 2 that does not divide c, since there must be another prime divisor q > 2. (End)

Numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016

Numbers 1 <= k <= n such that k | n^e with e >= 0. - Michael De Vlieger, May 29 2018

Examples

			n =  6: {1, 2, 3, 4, 6}.
n =  7: {1, 7}.
n =  8: {1, 2, 4, 8}.
n =  9: {1, 3, 9}.
n = 10: {1, 2, 4, 5, 8, 10}.
n = 11: {1, 11}.
n = 12: {1, 2, 3, 4, 6, 8, 9, 12}.
		

Links

• T. D. Noe and Michael De Vlieger, Rows n = 1..1000 of triangle, flattened (first rows n=1..200 from T. D. Noe)
• Michael De Vlieger, Plot k in row n at (x,y) = (k,-n) for n = 1..2^10.

Crossrefs

Cf. A007947, A010846 (number of terms in row n), A027750 (terms k that divide n), A243103 (product of terms in row n), A244974 (sum of terms in row n), A272618 (terms k that do not divide n).

Programs

Maple

A:= proc(n) local F, S, s, j, p;
  F:= numtheory:-factorset(n);
  S:= {1};
  for p in F do
    S:= {seq(seq(s*p^j, j=0..floor(log[p](n/s))), s=S)}
  od;
  S
end proc; map(op,[seq(A(n), n=1..100)]); # Robert Israel, Jul 15 2014

Mathematica

pf[n_] := If[n==1, {}, Transpose[FactorInteger[n]][[1]]]; SubsetQ[lst1_, lst2_] := Intersection[lst1,lst2]==lst1; Flatten[Table[pfn=pf[n]; Select[Range[n], SubsetQ[pf[ # ],pfn] &], {n,27}]]
(* Second program: *)
f[x_, y_ : 0] :=
  Block[{m, n, nn, j, k, p, t, v, z},
    n = Abs[x]; nn = If[y == 0, n, y];
    If[n == 1, {1},
      z = Length@
        MapIndexed[Set[{p[#2], m[#2]}, {#1, 0}] & @@
        {#1, First[#2]} &, FactorInteger[n][[All, 1]] ];
    k = Times @@ Array[p[#]^m[#] &, z]; Set[{v, t}, {1, False}];
    Union@ Reap[Do[Set[t, k > nn];
      If[t, k /= p[v]^m[v]; m[v] = 0; v++; If[v > z, Break[]],
      v = 1; Sow[k] ]; m[v]++; k *= p[v], {i, Infinity}] ][[-1, 1]] ] ];
Array[f, 120] (* Michael De Vlieger, Jun 18 2024 *)

Formula

Union of A027750 and nonzero terms of A272618.

Row n of this sequence is {k <= n : rad(k) | n }, where rad = A007947. - Michael De Vlieger, Jun 18 2024