cp's OEIS Frontend

A301415 Number of terms m in A002110 such that A301413(k) * A002110(m) is in A002182.

%I A301415 #12 May 16 2018 12:02:52
%S A301415 3,3,3,3,3,3,4,3,3,5,3,4,4,5,5,5,4,3,4,4,4,6,3,4,5,4,3,4,3,7,5,5,6,9,
%T A301415 6,5,8,6,8,8,8,6,6,8,6,5,7,8,9,5,5,7,6,5,6,5,6,5,6,9,9,6,9,9,6,6,7,8,
%U A301415 7,7,7,9,5,10,10,5,13,9,9,8,10,10,7,10,8
%N A301415 Number of terms m in A002110 such that A301413(k) * A002110(m) is in A002182.
%C A301415 Numbers m = A301414(x) * A002110(y) that are in A002182 are plotted below. Those also in A002201 are followed by asterisk.
%C A301415 This sequence counts the terms in each column.
%C A301415       1     2     3     4     5     6      7 ...
%C A301415   +-----------------------------------------
%C A301415 0 |   1
%C A301415 1 |   2*    4
%C A301415 2 |   6*   12*   24    36    48
%C A301415 3 |        60*  120*  180   240   360*   720
%C A301415 4 |             840  1260  1680  2520*  5040*
%C A301415 5 |                             27720  55440*
%C A301415 6 |                                   720720*
%C A301415 ...
%H A301415 Michael De Vlieger, <a href="/A301415/b301415.txt">Table of n, a(n) for n = 1..8000</a>
%H A301415 Michael De Vlieger, <a href="/A301415/a301415.png">Plot m = A301414(x) * A002110(y) at (x,y) for all 779674 m in Achim Flammenkamp's dataset</a> SHCNs are shown in red; 8563 * 4096 pixels.
%e A301415 a(1) = 3 since A301414(1) = 1 produces 3 highly composite numbers when multiplied by primorials p_0#, p_1#, and p_2# = {1, 2, 6}.
%e A301415 a(2) = 3 since A301414(2) = 2 yields 3 HCNs, multiplied by p_1#, p_2#, and p_3# = {4, 12, 60}.
%t A301415 f[n_] := With[{d = FactorInteger@ n}, If[n == 1, {0}, ReplacePart[Table[0, {PrimePi[d[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, d]]]; Take[#, 85] &@ Block[{s = a002182, a, b, c, m, u}, s = Take[s, 1000]; a = Array[{#2, #1, StringTrim[StringReplace[ToString@ #, ", " -> "."], ("{" | "}") ...] &[#3 /. {} -> 0], Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3]} & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {s[[#]], f@ s[[#]]} &, Length@ s]; u = Union@ a[[All, -1]]; b = MapIndexed[{i_, j_, k_, #1} -> ToExpression@ StringJoin["{i,", ToString@ First@ #2, ",", " j, k}"] &, Union@ a[[All, -1]]]; c = Map[# /. b &, a]; m = Max[c[[All, 2]] ]; c = Map[Sort@ # &, SplitBy[SortBy[c, First], First]]; Total /@ Transpose@ Array[With[{t = ConstantArray[0, m]}, ReplacePart[t, Map[#2 -> 1 & @@ # &, c[[#]] ] ] ] &, Length@ c] ]
%Y A301415 Cf. A002110, A002182, A301413, A301414.
%K A301415 nonn
%O A301415 1,1
%A A301415 _Michael De Vlieger_, Apr 09 2018