A002201 -id:A002201 - cp's OEIS frontend

A094348 Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.

1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 360360, 498960, 554400, 665280, 720720

Offset: 1

Matthew Vandermast, Jun 18 2004, Oct 12 2008

nonn

Includes all highly composite numbers (A002182) and least common multiples of 1 through n (A003418). It would be interesting to know: 1) whether or not all deeply composite numbers (A095848, which includes all members of A003418) also belong to this sequence; 2) if 72 is the only member of this sequence not also belonging to A002182 or A095848.
465585120 is the first member of A095848 that is not a member of this sequence. The first members that belong to neither A002182 nor A095848 are 72, 30240, 64864800 and 1470268800. - David Wasserman, Jun 28 2007
This sequence is also A096179 with duplicates deleted and sorted. Let F(n) be the number of the row of A096179 which has the first occurrence of a(n) and M(n) = max{F(i),i <= n}. Then the following table indicates this connection.
  n |1,2,3,4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17,  18,  19,  20, 21
a(n)|1,2,4,6,12,24,36,48,60,72,120,180,240,360,420,720,840,1260,1680,2520,5040
F(n)|1,2,4,3, 4, 8,18,16, 5, 9,  8, 18, 16,  9,  7, 16, 15,  21,  16,   9, 16
M(n)|1,2,4,4, 4, 8,18,18,18,18, 18, 18, 18, 18, 18, 18, 18,  21,  21,  21, 21
- Peter Luschny, Dec 29 2010

			72 is a multiple of seven of the first nine positive integers (namely, 1, 2, 3, 4, 6, 8 and 9). It is the smallest positive integer for which this is true.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 844.
J. Britton, Perfect Number Analyzer
Wikipedia, Table of divisors.
Peter Luschny, More terms, OEIS wiki.

Cf. A096179, A003418, A002182, A002201, A072938, A106037, A002110.

\\ Computes the first 50 terms of A094348
A094348() = {local(n,i,R,A,len,count,change,high,lim);
lim = 7208000; R = vector(500); A = vector(50); A[1]=1;
A[2]=2; A[3]=4; A[4]=6; A[5]=12; count=5; high=0; n=12;
while(n < lim, d=divisors(n); len=length(d); change=0;
for(i=1,min(len,high),if(R[i]>d[i],R[i]=d[i];change=1));
if(len>high,for(i=high+1,len,R[i]=d[i]); high=len);
if(change, count++; A[count] = n); n += 12; );
write("A094348.txt", vector(count, i, A[i])); }
\\ Peter Luschny, Dec 29 2010

More terms from David Wasserman, Jun 28 2007

Title edited by Matthew Vandermast, Nov 20 2010

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.

Original entry on oeis.org

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

Michael De Vlieger, Apr 09 2018

nonn

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.

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

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

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

(* 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]

A072938 Highly composite numbers that are half of the next highly composite number.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 2520

Offset: 1

Anonymous, Aug 12 2002

A002182(k) such that A002182(k+1) = 2*A002182(k).
Numbers m such that d(m)>=d(k) for 0A000005(n)}. - Lekraj Beedassy, Dec 16 2004
W. Brefeld (cf. link) gives a proof that there are no other terms. - Klaus Brockhaus, Mar 05 2006

			6 = A002182(4) is a term since A002182(5) = 12 = 2*6.

Werner Brefeld, Ramanujan, zusammengesetzte Zahlen und ihre Teilbarkeit.
Achim Flammenkamp, Highly Composite Numbers.

Cf. A002110, A002182, A002201, A003418, A094348, A106037, A165412.

Edited by Klaus Brockhaus, Aug 13 2002

A106037 Special highly composite numbers: a highly composite number (A002182) that divides all larger highly composite numbers.

Original entry on oeis.org

1, 2, 6, 12, 60, 2520

Offset: 1

N. J. A. Sloane, May 27 2005

J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988.

Shyam Sunder Gupta, Equal Product of Reversible Numbers (EPRN), Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 12, 353-365.
Steven Ratering, An interesting subset of the highly composite numbers, Math. Mag., 64 (1991), 343-346.

Cf. A002182, A002201, A000005.

Cf. A072938.

A166981 Superabundant numbers (A004394) that are highly composite (A002182).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800

Offset: 1

T. D. Noe, Oct 26 2009

The intersection of superabundant and highly composite numbers has exactly 449 terms, the largest of which is 2^10 * 3^6 * 5^4 * 7^3 * 11^3 * 13^2 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37*...*347.
The argument showing that this is a finite sequence seems to be given in A166735. - N. J. A. Sloane, Jan 04 2019
Pillai proved that this sequence is finite and asked for its number of terms (he used the term "highly abundant" for superabundant numbers). - Amiram Eldar, Jun 30 2019
From Michael De Vlieger, Dec 29 2020: (Start)
All terms are products of primorials (numbers in A002110), thus, all terms are also in A025487, itself a subsequence of A055932.
Since the colossally abundant numbers (CA, A004490) are also superabundant, and since the superior highly composite (SHC A002201) numbers are also highly composite, the finite sequence A224078 containing numbers both CA and SHC is a subsequence of this sequence. Likewise, A304234 (numbers that are SA, HC, & SHC but not CA) and A304235 (numbers that are SA, HC, & CA but not SHC), and A338786 (SA and HC, but neither CA nor SHC) are mutually exclusive finite subsequences of this sequence. (End)

T. D. Noe, Table of n, a(n) for n = 1..449 (complete sequence)
Michael De Vlieger, Annotated plot of a(n) at (x,y) = (a(n)/P), P) where P = A002110(A001221(a(n)) showing all 449 terms.
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. Mentions this sequence.
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.

Cf. A002110, A002182, A004394, A025487, A055932, A166735 (SA numbers that are not HC numbers), A224078, A304234, A304235, A308913, A338786.

A224078 Numbers that are superior highly composite and colossally abundant.

Original entry on oeis.org

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 321253732800, 6064949221531200, 791245405339403414400, 37188534050951960476800, 581442729886633902054768000

Offset: 1

David S. Metzler, Mar 30 2013

This should be all of the terms. Intersection of {highly composite} and {superabundant} has 449 terms, cf. A002182. The shapes (pattern of exponents of prime factors) of {superior highly composite} and {colossally abundant} seem to diverge for good after the last term listed here.

Finite intersection of A002201 and A004490.

A301413 a(n) = A002182(n)/A002110(A108602(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 12, 216, 240, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 72, 1440, 120, 144, 216, 240, 288, 360, 480, 576

Offset: 1

Michael De Vlieger, Mar 30 2018

nonn

This sequence appears in Siano paper, page 5 of 12, as the "variable part" v. - Michael De Vlieger, Oct 11 2023

			Let m be a value in this sequence. The table below shows m*A002110(A108602(k)). Columns are A108602(k), rows are m whose products m*A002110(A108602(k)) appear in A002182 are in this sequence. Numbers in A002182 that also appear in A002201 are followed by (*).
        0  1   2    3     4       5       6 ...
      +------------------------------------
    1 | 1* 2*  6*
    2 |    4  12*  60*
    4 |       24  120*  840
    6 |       36  180  1260
    8 |       48  240  1680
   12 |           360* 2520*  27720
   24 |           720  5040*  55440* 720720*
   ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, On a graph of highly composite numbers
A. Flammenkamp, Highly composite numbers
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.

Cf. A002110, A002182, A108602, A301414.

(* Load b-file from A002182 *)
With[{s = Import["b002182.txt","Data"][[All,-1]]}, Array[#/Product[Prime@ i, {i, PrimeNu[#]}] &@ s[[#]] &, 62]]

a(n) = A002182(n)/A007947(A002182(n)).

A059992 Numbers with an increasing number of nonprime divisors.

Original entry on oeis.org

1, 4, 8, 12, 24, 36, 48, 60, 72, 120, 180, 240, 360, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 75600, 83160, 110880, 151200, 166320, 221760, 277200, 332640

Offset: 1

Robert G. Wilson v, Mar 08 2001

nonn

Positions of records in A033273.
From Michael De Vlieger, Jan 04 2025: (Start)
Conjecture: This sequence includes all highly composite numbers (from A002182) except 2 and 6, but there are other terms in this sequence (e.g., a(3) = 8, a(9) = 72) that are not highly composite.
Conjecture: a(n)/A007947(a(n)) is in A301414. (End)

			a(4)=12 because twelve has 4 nonprime divisors {1, 4, 6 and 12} whereas a(3)=8 has only 3; and twelve is the first number greater than eight which exhibits this property.

Amiram Eldar, Table of n, a(n) for n = 1..571 (terms 1..146 from Ray Chandler)
Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.

Cf. A055079, A002182, A033273, A180040.

l = 0; Do[ c = Count[PrimeQ[ Divisors[n] ], False]; If[c > l, l = c; Print[n] ], {n, 1, 10^6} ]

lista(nn) = {my(m=0, nb); for (n=1, nn, nb = sumdiv(n, d, !isprime(d)); if (nb > m, m = nb; print1(n, ", ")););} \\ Michel Marcus, Jul 16 2019

Alternate description and b-file from Ray Chandler, Aug 07 2010

A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

Original entry on oeis.org

13967553600, 2248776129600, 65214507758400, 195643523275200, 12129898443062400, 448806242393308800, 18401055938125660800, 185942670254759802384000, 9854961523502269526352000, 1162885459773267804109536000, 780296143507862696557498656000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A002201 that are also in A004394 but not A004490.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304235.
There are 39 terms in this sequence.
The smallest term is 2^5 * 3^2 * 5 * A002110(8) or the product of A002110(k) with k = {1,1,1,2,3,8}.
The largest is 2^10 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(65) or the product of A002110(k) with k = {1,1,1,1,2,2,2,3,4,9,65}, a 144 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..39
Michael De Vlieger, Superior highly composite numbers m that are also superabundant but not colossally abundant
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark blue.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 39 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304235.

(* First, download b-files at A002201, A004394, and A004490 *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, _?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]};
With[{s = Import["b002201.txt", "Data"][[All, -1]], t = Select[Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ], IntegerQ@ First@ # &][[All, -1]], u = Import["b004490.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

A304235 Colossally abundant numbers that are highly composite, but not superior highly composite.

Original entry on oeis.org

160626866400, 9316358251200, 288807105787200, 2021649740510400, 224403121196654400, 9200527969062830400, 395622702669701707200, 1970992304700453905270400, 35468006523084668025340848000, 135483209545341953934626770390608000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A004490 that are also in A002182 but not A002201.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304234.
There are 32 terms in this sequence.
The smallest term is 2^4 * 3^2 * 5 * A002110(9) or the product of k = {1,1,2,3,9} in A002110.
The largest term is 2^9 * 3^5 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(66) or the product of A002110(k) with k = {1,1,1,1,2,2,3,4,9,66}, a 146 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..32
Michael De Vlieger, Colossally abundant m that are also highly composite but not superior highly composite
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark red.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 32 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304234.

(* First, download b-files at A002182, A002201, and A004490 *)
With[{s = Import["b004490.txt", "Data"][[All, -1]], t = Import["b002182.txt", "Data"][[All, -1]], u = Import["b002201.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

cp's OEIS Frontend

A094348 Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.

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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.

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A072938 Highly composite numbers that are half of the next highly composite number.

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A106037 Special highly composite numbers: a highly composite number (A002182) that divides all larger highly composite numbers.

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A166981 Superabundant numbers (A004394) that are highly composite (A002182).

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A224078 Numbers that are superior highly composite and colossally abundant.

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A301413 a(n) = A002182(n)/A002110(A108602(n)).

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A059992 Numbers with an increasing number of nonprime divisors.

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A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

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