The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A162306 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A162306

Irregular triangle in which row n contains the numbers <= n whose prime factors are a subset of prime factors of n.
(history; published version)

#41 by Peter Luschny at Thu Jun 20 02:34:04 EDT 2024

STATUS

reviewed

approved

#40 by Michel Marcus at Thu Jun 20 00:58:34 EDT 2024

STATUS

proposed

reviewed

#39 by Michael De Vlieger at Tue Jun 18 11:42:10 EDT 2024

STATUS

editing

proposed

#38 by Michael De Vlieger at Tue Jun 18 11:41:50 EDT 2024

MATHEMATICA

(* Second program: *)f[x_, y_ : 0] :=

f[x_, y_ : 0] :=

v = 1; Sow[k] ]; m[v]++; k *= p[v], {i, Infinity}] ][[-1, 1]] ] ]; Array[f, 120] (* _Michael De Vlieger_, Jun 18 2024 *)

Array[f, 120] (* Michael De Vlieger, Jun 18 2024 *)

Discussion

Tue Jun 18

11:42

Michael De Vlieger: Faster program.

#37 by Michael De Vlieger at Tue Jun 18 11:41:09 EDT 2024

LINKS

Michael De Vlieger, <a href="/A162306/a162306.png">Plot k in row n at (x,y) = (k,-n)</a> for n = 1..2^10.

FORMULA

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

MATHEMATICA

(* Second program: *)f[x_, y_ : 0] :=

Table[Select[Range@ n, PowerMod[n, Floor@ Log2@ n, #] == 0 &], {n,

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[]],

23v = 1; Sow[k] ]; m[v]++; k *= p[v], {i, Infinity}] // Flatten ][[-1, 1]] ] ]; Array[f, 120] (* Michael De Vlieger, May 29 2018 Jun 18 2024 *)

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).

STATUS

approved

editing

#36 by Susanna Cuyler at Mon Mar 30 20:59:20 EDT 2020

STATUS

reviewed

approved

#35 by Joerg Arndt at Mon Mar 30 11:13:05 EDT 2020

STATUS

proposed

reviewed

#34 by Michel Marcus at Mon Mar 30 09:24:04 EDT 2020

STATUS

editing

proposed

#33 by Michel Marcus at Mon Mar 30 09:23:59 EDT 2020

EXAMPLE

n = 6, a(n) = : {1, 2, 3, 4, 6}.

n = 7, a(n) = : {1, 7}.

n = 8, a(n) = : {1, 2, 4, 8}.

n = 9, a(n) = : {1, 3, 9}.

n = 10, a(n) = : {1, 2, 4, 5, 8, 10}.

n = 11, a(n) = : {1, 11}.

n = 12, a(n) = : {1, 2, 3, 4, 6, 8, 9, 12}.

STATUS

approved

editing

#32 by Susanna Cuyler at Tue May 29 20:38:44 EDT 2018

STATUS

proposed

approved