cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A181804 List of numbers that are LCMs of some set of highly composite numbers (A002182).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120, 144, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 60480, 75600, 83160, 90720, 100800, 110880, 151200, 166320, 181440, 221760, 226800, 277200
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

Numbers n such that A181801(n) is higher than A181801(d) for any proper divisor d of n. Also, numbers n such that row n of A181802 is identical to no previous row of A181802.
A002182 is a proper subsequence of this sequence. 72 is the first LCM of some set of highly composite numbers that is not itself highly composite.

Examples

			1, 2, 4, 6, 12, 24 and 36 are all highly composite numbers, and their least common multiple (LCM) is 72.  Hence, 72 is a member of the sequence.

Links

Crossrefs

A181805 gives the number of highly composite divisors of a(n), or A181801(a(n)).
Subsequence of A025487.
Includes all members of A181806.
Cf. A002182, A181802.

Programs

  • Mathematica
    seq[max_] := Module[{hcn = {}, hcnmax, d, dm = 0, s = {1}}, Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hcn, n]], {n, 1, max}]; hcnmax = hcn[[-1]]; Do[s = Union[Join[s, Select[LCM[hcn[[k]], s], # <= hcnmax &]]], {k, 2, Length[hcn]}]; s]; seq[300000] (* Amiram Eldar, Jun 23 2023 *)

A112780 Number of highly composite numbers (definition 1, A002182) with n decimal digits.

Original entry on oeis.org

4, 5, 6, 5, 9, 9, 9, 9, 10, 10, 10, 9, 11, 11, 8, 10, 11, 10, 11, 10, 9, 10, 13, 10, 12, 10, 13, 13, 13, 12, 13, 11, 14, 13, 13, 12, 13, 15, 13, 14, 13, 14, 13, 13, 15, 12, 14, 13, 17, 14, 16, 16, 15, 17, 15, 19, 15, 18, 15, 16, 17, 16, 17, 16, 15, 19, 15, 19, 14, 18, 14, 19, 17
Offset: 1

Views

Author

Ray Chandler, Nov 11 2005

Keywords

Examples

			a(1) = 4 since there are four highly composite numbers with one decimal digit {1,2,4,6}.

Links

Crossrefs

Cf. A002182, A002183, A108602, A112778, A112779, A112781.

Formula

First differences of A112781. - Amiram Eldar, Jul 02 2019

A181805 Number of divisors of A181804(n) that are highly composite (A002182).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 6, 7, 6, 7, 8, 8, 8, 10, 11, 14, 9, 9, 12, 14, 19, 15, 20, 21, 21, 20, 15, 22, 22, 22, 21, 23, 22, 17, 23, 23, 23, 24, 25, 24, 25, 23, 23, 25, 28, 25, 27, 27, 31, 22, 27, 26, 30, 18, 29, 25, 32, 33, 28, 29, 28, 35, 25, 33, 34, 31, 31, 38, 37
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

a(n) = maximal number of members of A002182 that have a least common multiple of A181804(n). Also, a(n) = length of row A181804(n) in triangles A181802 and A181803.
4, 13 and 16 are the first three positive integers that appear nowhere in this sequence (and, therefore, nowhere in A181801). It would be interesting to know whether there are others.

Examples

			A181804(10) = 72 has exactly seven divisors that are members of A002182 (namely, 1, 2, 4, 6, 12, 24 and 36). Hence, a(10) = 7.

Links

Crossrefs

A181806(m) is the m-th member of A181804 such that the value of a(n) increases to a record. See also A181807.
Cf. A002182, A181801, A181802, A181803, A181804.

Programs

Formula

a(n) = A181801(A181804(n)).

Extensions

More terms from Amiram Eldar, Jun 23 2023

A181809 Numbers n such that both n and n/2 are highly composite (A002182).

Original entry on oeis.org

2, 4, 12, 24, 48, 120, 240, 360, 720, 1680, 2520, 5040, 10080, 15120, 20160, 50400, 55440, 110880, 166320, 221760, 332640, 554400, 665280, 1441440, 2162160, 2882880, 4324320, 7207200, 8648640, 14414400, 17297280, 21621600, 43243200, 73513440
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

These are the numbers that set records both for total number of divisors and for number of even divisors; intersection of A002182 and A181808.
For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182).

Examples

			The number 12 is both highly composite (A002182(5) = 12) and twice another highly composite number (12 = 2*6 = 2*A002182(4)).  It therefore has more divisors (A002183(5)=6) than any smaller positive integer, and more even divisors (A002183(4)=4) than any smaller positive integer. Since 12 is the third positive integer with the properties that define this sequence, a(3)=12.

Links

Crossrefs

Numbers n such that 1 and 2 both appear in row n of A181803. See also A181808, A181810.
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors. Only the number 1 sets a record for its number of odd divisors and its number of total divisors.
Subsequence of A025487.

A212182 Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists exponents of distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 3, 2, 1, 4, 2, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 4, 2, 1, 1, 3, 3, 1, 1, 5, 2, 1, 1, 4, 3, 1, 1, 6, 2, 1, 1, 4, 2, 2, 1, 3, 2, 1, 1, 1, 4, 4, 1, 1, 5, 2, 2, 1, 4, 2, 1
Offset: 1

Views

Author

Matthew Vandermast, Jun 08 2012

Keywords

Comments

Length of row n = A108602(n).
For n > 1, row n of table gives the "nonincreasing order" version of the prime signature of A002182(n) (cf. A212171). This order is also the natural order of the exponents in the prime factorization of any highly composite number.
The distinct prime factors corresponding to exponents in row n are given by A318490(n, k), where k = 1, 2, 3, ..., A108602(n).

Examples

			First rows read:
  0;
  1;
  2;
  1, 1;
  2, 1;
  3, 1;
  2, 2;
  4, 1;
  2, 1, 1;
  3, 1, 1;
  2, 2, 1;
  4, 1, 1;
  ...
1st row: A002182(1) = 1 so T(1, 1) = 0;
2nd row: A002182(2) = 2^1 so T(2, 1) = 1;
3rd row: A002182(3) = 4 = 2^2 so T(3, 1) = 2;
4th row: A002182(4) = 6 = 2^1 * 3^1 so T(4, 1) = 1 and T(4, 2) = 1;
5th row: A002182(5) = 12 = 2^2 * 3^1 so T(5, 1) = 2 and T(5, 2) = 1;
6th row: A002182(6) = 24 = 2^3 * 3^1 so T(6, 1) = 3 and T(6, 2) = 1.

References

  • S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Links

Crossrefs

Row n has length A108602(n), n >= 2.
Cf. A000040, A002182, A124010, A212171, A318490.

Formula

Row n equals row A002182(n) of table A124010. For n > 1, row n equals row A002182(n) of table A212171.

Extensions

Edited by Peter J. Marko, Aug 30 2018

A306587 Numbers k such that A002182(k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 18, 20, 22, 23, 26, 28, 30, 34, 35, 44, 49, 54, 57, 60, 63, 74, 78, 84, 91, 97, 102, 104, 108, 111, 112, 114, 116, 118, 126, 134, 143, 145, 149, 159, 162, 167, 173, 177, 179, 188, 204, 208, 214, 230, 236, 238, 247, 249, 280, 294, 298
Offset: 1

Views

Author

Dmitry Kamenetsky, Mar 02 2019

Keywords

Links

Crossrefs

Cf. A002182 (highly composite numbers), A072828, A306588.

Extensions

More terms from Daniel Suteu, Mar 02 2019

A306588 Numbers k such that A002182(k)-1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 20, 21, 28, 30, 31, 37, 39, 40, 45, 52, 55, 65, 66, 67, 68, 70, 79, 81, 84, 101, 108, 118, 131, 132, 136, 143, 148, 149, 151, 163, 170, 174, 185, 191, 200, 203, 208, 212, 231, 259, 261, 286, 289, 297, 317, 326, 327
Offset: 1

Views

Author

Dmitry Kamenetsky, Mar 02 2019

Keywords

Links

Crossrefs

Cf. A002182 (highly composite numbers), A072826, A306587.

Extensions

More terms from Daniel Suteu, Mar 02 2019

A324582 a(n) = A002182(n) * A324581(n) = A002182(n) * A276086(A002182(n)).

Original entry on oeis.org

2, 6, 36, 30, 300, 15000, 1260, 42000, 2940, 288120, 21176820, 18480, 66555720, 328703760, 12298440, 2232166860, 360122920080, 360360, 103062960, 22107004920, 4215068938080, 129290917072196880, 3525159950945805332160, 90107494796113466546674800, 645822919595173320, 72532204477502449680, 1648012277067163992784800
Offset: 1

Views

Author

Antti Karttunen, Mar 09 2019

Keywords

Comments

Note that gcd(A002182(n), A324581(n)) = A324198(A002182(n)) = 1 for all n because each term of A002182 is a product of primorial numbers (A002110).
See also comments in A324382.

Links

Crossrefs

Subsequence of A324577.
Cf. A002110, A002182, A276086, A324198, A324382, A324580, A324581.

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20], s = DivisorSigma[0, Range[10^5]], t}, t = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Array[#1 (Times @@ Power @@@ Transpose@ {Prime@ Range@ Length@ #2, Reverse@ #2}) & @@ {#, IntegerDigits[#, b]} &@ t[[#]] &, Length@ t]] (* Michael De Vlieger, Mar 18 2019 *)
  • PARI
    \\ A002182 assumed to be precomputed
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324582(n) = A002182(n)*A276086(A002182(n));

Formula

a(n) = A002182(n) * A324581(n) = A002182(n) * A276086(A002182(n)).
a(n) = A324580(A002182(n)).

A063072 Sum of divisors of Ramanujan's highly composite numbers, or sigma(A002182(n)).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 360, 546, 744, 1170, 2418, 2880, 4368, 5952, 9360, 19344, 28800, 39312, 59520, 79248, 99944, 112320, 180048, 203112, 232128, 345600, 471744, 714240, 950976, 1199328, 1451520, 2160576, 2437344, 2926080
Offset: 1

Views

Author

Jason Earls, Aug 02 2001

Keywords

Links

Crossrefs

Cf. A000005, A004394, A002182, A002183, A002473, A000203.

Programs

  • Mathematica
    s={}; dm=0; Do[d = DivisorSigma[0,n]; If[d > dm, dm=d; AppendTo[s, DivisorSigma[1,n]]], {n, 1, 10^5}]; s (* Amiram Eldar, Jun 28 2019 *)
  • PARI
    a=0; j=[]; for(n=1,200000,b=numdiv(n); if(b>a,a=b; j=concat(j, sigma(n)))); j
  • PARI
    { n=a=0; for (m=1, 10^9, b=numdiv(m); if(b>a, a=b; write("b063072.txt", n++, " ", sigma(m)); if (n==50, break)) ) } \\ Harry J. Smith, Aug 16 2009

Formula

a(n) = A000203(A002182(n)). - Michel Marcus, Jun 28 2018

Extensions

More terms from Reiner Martin, Dec 22 2001

A181807 Number of divisors of A181806(n) that are highly composite (A002182).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 14, 19, 20, 21, 22, 23, 24, 25, 28, 31, 32, 33, 35, 38, 39, 41, 48, 49, 52, 53, 57, 59, 65, 67, 69, 77, 81, 82, 86, 91, 94, 103, 105, 107, 114, 118, 122, 125, 131, 132, 135, 141, 142, 144, 145, 154, 157, 160, 163, 166, 171, 175, 180
Offset: 1

Views

Author

Matthew Vandermast, Nov 27 2010

Keywords

Comments

Also, length of row A181806(n) in triangles A181802 and A181803.

Examples

			A181806(4) = 12 has exactly five divisors (namely, 1, 2, 4, 6 and 12) that are members of A002182.  Hence, a(4) = 5.

Links

Crossrefs

Cf. A002182, A181801, A181802, A181803, A181806.

Formula

a(n) = A181801(A181806(n)).

Extensions

More terms from Amiram Eldar, Aug 29 2019 (calculated from the b-file at A181806)
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