cp's OEIS Frontend

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.

%I A301414 #23 Jun 30 2025 04:20:14
%S A301414 1,2,4,6,8,12,24,36,48,72,96,120,144,216,240,288,360,480,576,720,1080,
%T A301414 1440,2160,2880,4320,5040,7200,7560,8640,10080,14400,15120,20160,
%U A301414 30240,40320,50400,60480,90720,100800,120960,151200,181440,241920,302400,362880
%N 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.
%C A301414 Given that highly composite numbers (HCNs) are products of primorials, we note the following:
%C A301414 1. The only odd term is 1.
%C A301414 2. The only primorials, i.e., terms in A002110, are {1, 2, 6}, consequently the only squares in A002182 are {1, 4, 36}.
%C A301414 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.
%C A301414 4. This sequence is a subset of A025487, which is a subset of A055932.
%C A301414 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.
%C A301414 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.
%C A301414 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.
%H A301414 Michael De Vlieger, <a href="/A301414/b301414.txt">Table of n, a(n) for n = 1..8000</a>
%H A301414 Michael De Vlieger, <a href="http://vincico.com/proof/A301414.html">On a graph of highly composite numbers</a>
%H A301414 Michael De Vlieger, <a href="/A301414/a301414.png">Plot of 779674 HCNs as A002110(y) * A301414(x) at (x,y)</a>.
%H A301414 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly composite numbers</a>
%e A301414 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.
%e A301414    {0,1} {1,1} {2,1}
%e A301414      1     2*    6*
%e A301414          {1,2} {2,2} {3,2}
%e A301414            4     12*   60*
%e A301414                {2,4} {3,4}  {4,4}
%e A301414                  24   120*   840
%e A301414                {2,6} {3,6}  {4,6}
%e A301414                  36   180    1260
%e A301414                {2,8} {3,8}  {4,8}
%e A301414                  48   240    1680
%e A301414                     {3,12} {4,12}   {5,12}
%e A301414                       360*   2520*   27720
%e A301414                     {3,24} {4,24}   {5,24}    {6,24}
%e A301414                       720    5040*   55440*   720720*
%e A301414                            {4,36}   {5,36}    {6,36}
%e A301414                              7560    83160   1081080
%e A301414                            {4,48}   {5,48}    {6,48}
%e A301414                             10080   110880   1441440*
%e A301414                             ...     ...      ...       ...
%t A301414 (* First load b-file from A002182 minus any comments therein *)
%t A301414 s = Import["b002182.txt","Data"][[All,-1]];
%t A301414 (* Alternatively, download Flammenkamp dataset, decompress and rename to "HCN.txt", then decode using the following in place of s above *)
%t A301414 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"];
%t A301414 Union@ Array[#1/Product[Prime@ i, {i, #2}] & @@ {#, PrimeNu@ #} &@ s[[#]] &, Length@ s]
%Y A301414 Cf. A000079, A002110, A002182, A002201, A025487, A055932, A275055.
%K A301414 nonn
%O A301414 1,2
%A A301414 _Michael De Vlieger_, Apr 09 2018