A301414 Distinct terms of A301413 in ascending order: terms k in A301413 that have at least one number m such that k * A002110(m) is a highly composite number (A002182) with m distinct prime factors.

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

Author

Michael De Vlieger, Apr 09 2018

Keywords

• nonn

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.

4. 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.

Examples

			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*
                            ...     ...      ...       ...
		

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

Crossrefs

Cf. A000079, A002110, A002182, A002201, A025487, A055932, A275055.

Programs

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]