A301414 Numbers k in A301413 such that k * A002110(m) is in A002182.

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2880, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 30240, 40320, 50400, 60480, 90720, 100800, 120960, 151200, 181440, 241920, 302400, 362880

Offset

1,2

Comments

Given that highly composite numbers (HCNs) are products of primorials, we note the following:

1. The only odd term is 1.

2. The only primorials, i.e., terms in A002110, are {1, 2, 6}, consequently the only squares in A002182 are {1, 4, 36}.

3. The only terms in A000079 are {1, 2, 4, 8}. These produce {1, 2, 6}, {4, 12, 30}, {24, 120, 840}, and {48, 240, 1680}, in A002182 respectively.

3. This sequence is a subset of A025487, which is a subset of A055932.

Also given that A002182 strictly increases, we note that i <= m <= j, integers, for which P = k * A002110(m) produces HCNs. As we increment m we increase the rank of the tensor of prime divisor power ranges and double the number of divisors. However, we may have another term P' = a * A002110(b) for a > k and b < (j + 1) such that P' < P yet tau(P') >= tau(P). This P' is in A002182 and has increased tau by the lengthening of the power ranges for relatively small primes via some composite b instead of increasing the rank of the tensor. Since A002182 strictly increases, we have a limited range for m.

There are 19 terms also in A002182: 1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 5040, 7560, 10080, 15120, 20160, 50400, 17297280.

Let n = A002110(m), and consider the ordered pair (n, k). In a plot of ordered pairs that produce m in A002182, we have the first terms of A002182 thus: (0,1), (1,1), (1,2), (2,1), (2,2), (2,4), (2,6), (2,8), (3,2), (3,4), (3,6), (3,8), (3,12), etc.

Links

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

Michael De Vlieger, On a graph of highly composite numbers

Michael De Vlieger, Plot of 779674 HCNs as A002110(y) * A301414(x) at (x,y).

A. Flammenkamp, Highly composite numbers

Example

Plot of (n,k) with n in A002110 and k a term in this sequence such that A002110(n) * k is in A002182. Asterisks denote products that are in A002201.
   {0,1} {1,1} {2,1}
     1     2*    6*
         {1,2} {2,2} {3,2}
           4     12*   60*
               {2,4} {3,4}  {4,4}
                 24   120*   840
               {2,6} {3,6}  {4,6}
                 36   180    1260
               {2,8} {3,8}  {4,8}
                 48   240    1680
                    {3,12} {4,12}   {5,12}
                      360*   2520*   27720
                    {3,24} {4,24}   {5,24}    {6,24}
                      720    5040*   55440*   720720*
                           {4,36}   {5,36}    {6,36}
                             7560    83160   1081080
                           {4,48}   {5,48}    {6,48}
                            10080   110880   1441440*
                            ...     ...      ...       ...

Mathematica

(* First load b-file from A002182 minus any comments therein *)
s = Import["b002182.txt", "Data"][[All, -1]];
(* Alternatively, download Flammenkamp dataset, decompress and rename to "HCN.txt", then decode using the following in place of s above *)
s = Times @@ {Times @@ Prime@ Range@ ToExpression@ First@ #1, If[# == {}, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, #]] &@ DeleteCases[-1 + Flatten@ Map[If[StringFreeQ[#, "^"], ToExpression@ #, ConstantArray[#1, #2] & @@ ToExpression@ StringSplit[#, "^"]] &, #2], 0]} & @@ TakeDrop[Drop[StringSplit@ #, 2], 1] & /@ Import["HCN.txt", "Data"];
Union@ Array[#1/Product[Prime@ i, {i, #2}] & @@ {#, PrimeNu@ #} &@ s[[#]] &, Length@ s]]

Crossrefs

Cf. A000079, A002110, A002182, A002201, A025847, A055932, A275055.

Sequence in context: A168267 A308912 A333952 * A333953 A333963 A307187

Adjacent sequences: A301411 A301412 A301413 * A301415 A301416 A301417

Keyword

nonn

Author

Michael De Vlieger, Apr 09 2018

Status

approved