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

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, 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, 7, 7, 7, 9, 5, 10, 10, 5, 13, 9, 9, 8, 10, 10, 7, 10, 8

Offset

1,1

Comments

Numbers m = A301414(x) * A002110(y) that are in A002182 are plotted below. Those also in A002201 are followed by asterisk.

This sequence counts the terms in each column.

1 2 3 4 5 6 7 ...

+-----------------------------------------

0 | 1

1 | 2* 4

2 | 6* 12* 24 36 48

3 | 60* 120* 180 240 360* 720

4 | 840 1260 1680 2520* 5040*

5 | 27720 55440*

6 | 720720*

...

Links

Michael De Vlieger, Table of n, a(n) for n = 1..8000

Michael De Vlieger, Plot m = A301414(x) * A002110(y) at (x,y) for all 779674 m in Achim Flammenkamp's dataset SHCNs are shown in red; 8563 * 4096 pixels.

Example

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}.
a(2) = 3 since A301414(2) = 2 yields 3 HCNs, multiplied by p_1#, p_2#, and p_3# = {4, 12, 60}.

Mathematica

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] ]

Crossrefs

Cf. A002110, A002182, A301413, A301414.

Sequence in context: A063438 A276870 A081168 * A210509 A362049 A333819

Adjacent sequences: A301412 A301413 A301414 * A301416 A301417 A301418

Keyword

nonn

Author

Michael De Vlieger, Apr 09 2018

Status

approved