A000705 -id:A000705 - cp's OEIS frontend

A002201 Superior highly composite numbers: positive integers n for which there is an e > 0 such that d(n)/n^e >= d(k)/k^e for all k > 1, where the function d(n) counts the divisors of n (A000005).

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200, 12129898443062400, 448806242393308800, 18401055938125660800, 791245405339403414400

Offset: 1

N. J. A. Sloane

For fixed e > 0, d(n)/n^e is bounded and reaches its maximum at one or more points.
This is an infinite subset of A002182.
The first 15 numbers in this sequence agree with those in A004490 (colossally abundant numbers). - David Terr, Sep 29 2004

			For n=2, 6 and 12 we may take e in the intervals (log(2)/log(3), 1], (log(3/2)/log(2), log(2)/log(3)] and (log(2)/log(5), log(3/2)/log(2)], respectively.
Can the intervals in the previous line can be extended to include the left endpoints? - _Ant King_, May 02 2005

J. L. Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988.
S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
S. Ramanujan, Highly Composite Numbers: Section IV, in 1) Collected Papers of Srinivasa Ramanujan, pp. 111-8, Ed. G. H. Hardy et al., AMS Chelsea 2000. 2) Ramanujan's Papers, pp. 143-150, Ed. B. J. Venkatachala et al., Prism Books Bangalore 2000.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iain Fox, Table of n, a(n) for n = 1..400 (first 150 terms from T. D. Noe)
Hirotaka Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259 [math.NT], 2024. See p. 4.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.
S. Ramanujan, IV: Superior Highly Composite Numbers
S. Ratering, An interesting subset of the highly composite numbers, Math. Mag., 64 (1991), 343-346.
Eric Weisstein's World of Mathematics, Superior Highly Composite Number
Eric Weisstein's World of Mathematics, Colossally Abundant Number
Wikipedia, Superior highly composite number

Cf. A000705, A004490, A000005.

Cf. A002182, A072938, A106037, A094348, A003418, A002110, A263572.

Rest@ Union@ Array[Product[p^Floor[1/(p^(1/#) - 1)], {p, Prime@ Range@ PrimePi[2^#]}] &[Log@ #] &, 160] (* Michael De Vlieger, Jul 09 2019 *)

lista(nn)=my(p=primes(primepi(2^log(nn)))); setminus(Set(vector(nn, i, prod(n=1, primepi(2^log(i)), p[n]^floor(1/(p[n]^(1/log(i))-1))))), [1]) \\ Iain Fox, Aug 23 2020

Better definition from T. D. Noe, Nov 05 2002

A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant.

Original entry on oeis.org

1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 10080, 15120, 25200, 27720, 110880, 166320, 277200, 332640, 554400, 665280, 2162160, 3603600, 7207200, 8648640, 10810800, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 698377680, 735134400, 1102701600

Offset: 1

Michael De Vlieger, Nov 09 2020

These are numbers both highly composite and superabundant but neither superior highly composite nor colossally abundant.
This sequence, A224078, A304234, and A304235 are mutually exclusive subsets that comprise A166981.
Superset A166981 has 449 terms; this sequence has 358, A224078 has 20, A304234 has 39, and A304235 has 32.

			1 is in the sequence since it is the empty product, setting records for both the number of divisors and the sum of divisors, and it is neither also superior highly composite nor colossally abundant.
2 is not in the sequence since it is both colossally abundant and superior highly composite.
4 is in the sequence since it sets a record for the divisor counting and divisor sum functions, yet it is neither superior highly composite nor colossally abundant.
20951330400 is not in the sequence since it is colossally abundant though it is an HCN and SA. etc.

Michael De Vlieger, Table of n, a(n) for n = 1..358
Michael De Vlieger, Annotated plot of a(n) at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))
Eric Weisstein's World of Mathematics, Highly Composite Number
Eric Weisstein's World of Mathematics, Colossally Abundant Number
Eric Weisstein's World of Mathematics, Superabundant Number
Eric Weisstein's World of Mathematics, Superior Highly Composite Number

Cf. A189228, A002201, A002182, A004394, A004490, A166981, A224078, A304234, A304235.

Complement[Import["https://oeis.org/A166981/b166981.txt", "Data"][[1 ;; 449, -1]], Union[FoldList[Times, Import["https://oeis.org/A073751/b073751.txt", "Data"][[1 ;; 120, -1]] ], FoldList[Times, Import["https://oeis.org/A000705/b000705.txt", "Data"][[1 ;; 120, -1]] ] ] ] (* Program reads OEIS b-files Michael De Vlieger, Nov 09 2020 *)

Complement of (the union of A002182 and A004394) and (the union of A002201 and A004490).

A301416 Numbers k in A301413 such that k * A002110 (m) is in A002201.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 144, 720, 1440, 10080, 30240, 60480, 302400, 604800, 6652800, 19958400, 259459200, 518918400, 3632428800, 61751289600, 185253868800, 926269344000, 17599117536000, 35198235072000, 809559406656000, 1619118813312000, 4857356439936000

Offset: 1

Michael De Vlieger, Apr 09 2018

nonn

			From _Michael De Vlieger_, May 14 2018: (Start)
Numbers m = A301416(x) * A002110(y) that are in A002201 are plotted below.
      1     2     3     4      5        6        7 ...
  +-----------------------------------------------
1 |   2
2 |   6    12
3 |        60   120   360
4 |                  2520   5040
5 |                        55440
6 |                       720720  1441440  4324320
...
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..170
Michael De Vlieger, Plot m = a(x) * A002110(y) at (x,y) for m in A002201, smallest 4096 terms m.

Cf. A002110, A002201, A301413, A301414.

t = Import["b002201.txt", "Data"][[All, -1]]; (* Uses b-file at A002201 Alternatively, use this conversion of terms at A000705 to a 10^4 term dataset for A002201. Processing 10^4 terms will take a long time: *) t = With[{s = Import["b000705.txt", "Data"][[All, -1]]}, FoldList[Times, s]]; f[n_] := With[{d = FactorInteger@ n}, If[n == 1, {0}, ReplacePart[Table[0, {PrimePi[d[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, d]]]; Union@ Array[Times @@ MapIndexed[Prime[First@#2]^#1 &, #3] & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {t[[#]], f@ t[[#]]} &, Length@ t]

A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 6, 12, 24, 72, 144, 288, 864, 1728, 5184, 10368, 20736, 62208, 124416, 373248, 746496, 1492992, 4478976, 8957952, 26873856, 53747712, 107495424, 322486272, 644972544, 1289945088, 3869835264, 7739670528, 23219011584, 46438023168, 92876046336, 278628139008, 557256278016

Offset: 1

Amiram Eldar, Jul 06 2019

nonn

How is this related to A163895? - R. J. Mathar, May 05 2023

			From _Michael De Vlieger_, Jul 12 2019: (Start)
We can plot all terms in A003586 with the power range 2^x with x >= 0 and 3^y with y >= 0 on the x and y axis, respectively. Plot of terms m in A309015, with terms also in a(n) placed in brackets:
                                2^x
          0    1     2     3     4     5     6     7     8
        +-----------------------------------------------------
     0  |[1]  [2]    4
     1  |     [6]  [12]  [24]   48
3^y  2  |           36   [72] [144]  [288]   576
     3  |                216   432   [864] [1728] 3456  6912 ...
          ...
Larger scale plot with "." representing a term m in A309015, and "o" representing a term in A309015 also in a(n) for all m < A002110(20).
                              2^x
        0    5   10   15   20   25   30   35   40   45  ...
        +------------------------------------------------
       0|oo.
        | ooo.
        |  .ooo.
        |   ..oo..
        |    ..ooo..
       5|      ..oo...
        |       ..ooo...
        |         ..oo....
        |          ..ooo....
        |            ..ooo....
      10|             ...oo.....
        |               ..ooo....
        |                ...oo.....
        |                  ..ooo.....
3^y     |                   ...ooo....
      15|                     ...oo.....
        |                      ...ooo.....
        |                        ...oo.....
        |                         ...ooo.....
        |                           ...oo......
      20|                            ...ooo.....
        |                              ...ooo.....
        |                               ....oo......
        |                                 ...ooo.....
        |                                  ....oo......
      25|                                    ...ooo......
        |                                     ....ooo....
        |                                       ....oo.
        |                                        ....o
        |                                          .
     ...
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..2709
Gérard Bessi, Etude des nombres 2-hautement composés, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.
Michael De Vlieger, Factors p analogous to A000705 such that the product of the smallest n terms equals a(n + 1) (10^5 terms).

Subsequence of A003586 and A309015.

Cf. A000005, A002201.

f[nn_, k_: 2] := Block[{w = {{2, 1}, {3, 0}}, s = {2}, P = 1, q = k - 2, x, i, n, f}, f[w_List] := Log[#1, (#2 + 2)/(#2 + 1)] & @@ w; x = Array[f[w[[#]] ] &, P + 1]; For[n = 2, n <= nn, n++, i = First@ FirstPosition[x, Max[x]]; AppendTo[s, w[[i, 1]]]; w[[i, 2]]++; If[And[i > P, P <= q], P++; AppendTo[w, {Prime[i + 1], 0}]; AppendTo[x, f[Last@ w]]]; x[[i]] = f@ w[[i]] ]; s]; {1}~Join~FoldList[Times, f[32, 2]] (* Michael De Vlieger, Jul 11 2019, after T. D. Noe at A000705 *)

More terms from Michael De Vlieger, Jul 11 2019

A370634 A135507(n) is the product of the first n terms of this sequence.

Original entry on oeis.org

1, 4, 5, 3, 3, 3, 9, 4, 3, 3, 13, 3, 3, 9, 3, 3, 19, 3, 3, 3, 9, 13, 25, 3, 3, 3, 3, 9, 31, 3, 3, 4, 13, 19, 9, 3, 39, 3, 3, 3, 43, 9, 3, 13, 3, 25, 49, 3, 3, 3, 19, 3, 55, 3, 3, 3, 3, 31, 61, 3, 3, 3, 3, 3, 3, 3, 69, 19, 3, 3, 73, 3, 3, 39, 3, 3, 3, 3, 81, 3

Offset: 1

Michael De Vlieger, May 19 2024

nonn

Compactification of A135507 akin to A000705 with respect to A002201.

			Table of first 20 terms of this sequence and S = A135507.
   n            S(n)  a(n)
  ------------------------
   1              1     1
   2              4     4
   3             20     5
   4             60     3
   5            180     3
   6            540     3
   7           4860     9
   8          19440     4
   9          58320     3
  10         174960     3
  11        2274480    13
  12        6823440     3
  13       20470320     3
  14      184232880     9
  15      552698640     3
  16     1658095920     3
  17    31503822480    19
  18    94511467440     3
  19   283534402320     3
  20   850603206960     3

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A001359, A003418, A006512, A135507.

nn = 120; j = 1; {1}~Join~Reap[Do[k = 2 j + LCM[j, i]; Sow[k/j]; j = k, {i, 2, nn}] ][[-1, 1]]

For n > 1, 3 <= a(n) <= n+2.

For p = A001359(i) such that gcd(a(p-1), p) = 1, a(p) = p+2 = A006512(i).

A098895 Number of divisors of the n-th superior highly composite number.

Original entry on oeis.org

2, 4, 6, 12, 16, 24, 48, 60, 120, 240, 288, 384, 576, 1152, 2304, 2688, 5376, 8064, 16128, 20160, 40320, 46080, 92160, 184320, 368640, 737280, 983040, 1966080, 3932160, 4423680, 6635520, 13271040, 15925248, 31850496, 63700992, 127401984

Offset: 1

David Terr, Oct 14 2004

nonn

Sequence A002201 gives the values of the n-th superior highly composite number N(n) and A000705 gives the values of the (prime) ratio N(n)/N(n-1).

			a(8)=60 because the eighth superior highly composite number, 5040, has 60 divisors.

S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.

Amiram Eldar, Table of n, a(n) for n = 1..3401 (calculated using the b-file at A000705)
S. Ramanujan, Highly Composite Numbers.

Cf. A000705, A002201.

a(n) = a(n-1) * (1 + 1/k(n)), where k(n) is the p(n)-adic valuation of the n-th superior highly composite number N(n), with p(n) = N(n)/N(n-1) and N(0)=1.

A098896 p(n)-adic valuation of the n-th superior highly composite number N(n), where p(n) = N(n)/N(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 1, 1, 5, 3, 2, 1, 1, 6, 1, 2, 1, 4, 1, 7, 1, 1, 1, 1, 3, 1, 1, 8, 2, 1, 5, 1, 1, 1, 1, 2, 1, 1, 9, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 6, 4, 1, 1, 2, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

David Terr, Oct 14 2004

nonn

(1+1/a(n)) appears in the denominators of the log arguments of the denominators of the numbers in the table of the reference, pp. 115-117.

			a(8) = 4 since N(8)=5040 has 2-adic valuation of 4 and N(8)/N(7)=2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using the b-file at A000705)
S. Ramanujan, Highly Composite Numbers.
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, pp. 78-129. See esp. pp. 87, 115-117.

Cf. A000705, A002201.

More terms from Amiram Eldar, Aug 12 2019

A189466 Number of superior highly composite numbers < 10^n.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41, 42, 42, 43, 43, 44, 44, 46, 46, 46, 47, 47, 48, 49, 49, 50, 50, 51, 51, 52, 52, 53

Offset: 1

Krzysztof Ostrowski, Apr 22 2011

nonn

Number of superior highly composite numbers (A002201) with at most n digits.

			a(2) = 4 since there are 4 superior highly composite numbers < 10^2 {2,6,12,60}

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using the b-file at A000705; terms 1..250 from Krzysztof Ostrowski)
T. D. Noe, List of superior highly composite numbers

Cf. A000705, A002201, A112781.

A328852 Quotients of consecutive terms of A328549.

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2

Offset: 1

Hal M. Switkay, Oct 28 2019

A328549(n+1) is the product of the first n terms of this sequence. By the definition of A328549, this sequence is an initial sequence of both A000705 and A073751.

Cf. A000705, A073751, A328549.

a(n) = A328549(n+1)/A328549(n).

cp's OEIS Frontend

A002201 Superior highly composite numbers: positive integers n for which there is an e > 0 such that d(n)/n^e >= d(k)/k^e for all k > 1, where the function d(n) counts the divisors of n (A000005).

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A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant.

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A301416 Numbers k in A301413 such that k * A002110 (m) is in A002201.

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A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

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A370634 A135507(n) is the product of the first n terms of this sequence.

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A098895 Number of divisors of the n-th superior highly composite number.

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A098896 p(n)-adic valuation of the n-th superior highly composite number N(n), where p(n) = N(n)/N(n-1).

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A189466 Number of superior highly composite numbers < 10^n.

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