A004394 -id:A004394 - cp's OEIS frontend

A362081 Numbers k achieving record abundance (sigma(k) > 2*k) via a residue-based measure M(k) (see Comments), analogous to superabundant numbers A004394.

1, 2, 4, 6, 12, 24, 30, 36, 72, 120, 360, 420, 840, 1680, 2520, 4032, 5040, 10080, 25200, 32760, 65520, 98280, 194040, 196560, 388080, 942480, 1801800, 3160080, 3603600, 6320160, 12640320, 24504480, 53721360, 61981920, 73513440, 115315200, 122522400, 189909720, 192099600, 214885440

Offset: 1

Richard Joseph Boland, Apr 08 2023

nonn

The residue-based quantifier function, M(k) = (k+1)*(1 - zeta(2)/2) - 1 - ( Sum_{j=1..k} k mod j )/k, measures either abundance (sigma(k) > 2*k), or deficiency (sigma(k) < 2*k), of a positive integer k. It follows from the known facts that Sum_{j=1..k} (sigma(j) + k mod j) = k^2 and that the average order of sigma(k)/k is Pi^2/6 = zeta(2) (see derivation below).

M(k) ~ 0 when sigma(k) ~ 2*k and for sufficiently large k, M(k) is positive when k is an abundant number (A005101) and negative when k is a deficient number (A005100). The terms of this sequence are the abundant k for which M(k) > M(m) for all m < k, analogous to the superabundant numbers A004394, which utilize sigma(k)/k as the measure. However, sigma(k)/k does not give a meaningful measure of deficiency, whereas M(k) does, thus a sensible notion of superdeficient (see A362082).

			The abundance measure is initially negative, becoming positive for k > 30. Initial measures with factorizations from the Mathematica program:
   1  -0.64493406684822643647   {{1,1}}
   2  -0.46740110027233965471   {{2,1}}
   4  -0.36233516712056609118   {{2,2}}
   6  -0.25726923396879252765   {{2,1},{3,1}}
  12  -0.10873810118013850374   {{2,2},{3,1}}
  24  -0.10334250226949712257   {{2,3},{3,1}}
  30  -0.096478036147509765322  {{2,1},{3,1},{5,1}}
  36   0.068719763307810925260  {{2,2},{3,2}}
  72   0.12657322670640173542   {{2,3},{3,2}}

Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemmann Hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Amer. Math. Monthly, 109 (2002), 534-543.

Cf. A362082 (superdeficient), A362083 (analogous to A335067, A326393).

Cf. A004394, A004490, A002201, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.

Clear[max, Rp, R, seqtable, M];
max = -1; Rp = 0; seqtable = {};
Do[R = Rp + 2 k - 1 - DivisorSigma[1, k];
  M = N[(k + 1)*(1 - Zeta[2]/2) - 1 - R/k, 20];
  If[M > max, max = M; Print[k, "   ", max, "   ", FactorInteger[k]];
   AppendTo[seqtable, k]];
  Rp = R, {k, 1, 1000000000}];
Print[seqtable]

M(n) = (n+1)*(1 - zeta(2)/2) - 1 - sum(k=2, n, n%k)/n;
lista(nn) = my(m=-oo, list=List()); for (n=1, nn, my(mm = M(n)); if (mm > m, listput(list, n); m = mm);); Vec(list); \\ Michel Marcus, Apr 21 2023

Derived starting with lemmas 1-3:
1) Sum_{j=1..k} (sigma(j) + k mod j) = k^2.
2) The average order of sigma(k)/k is Pi^2/6 = zeta(2).
3) R(k) = Sum_{j=1..k} k mod j, so R(k)/k is the average order of (k mod j).
Then:
Sum_{j=1..k} sigma(j) ~ zeta(2)*Sum_{j=1..k} j = zeta(2)*(k^2+k)/2.
R(k)/k ~ k - k*zeta(2)/2 - zeta(2)/2.
0 ~ (k+1)*(1 - zeta(2)/2) - 1 - R(k)/k.
Thus M(k) = (k+1)*(1 - zeta(2)/2) - 1 - R(k)/k is a measure of variance about sigma(k) ~ 2*k corresponding to M(k) ~ 0.

A073349 Numbers m such that sigma(m)/m is equal to sigma(k)/k for some k being superabundant (A004394).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 28, 36, 48, 60, 120, 180, 234, 240, 360, 496, 600, 672, 720, 840, 1170, 1260, 1680, 1800, 2016, 2520, 4200, 5040, 8128, 10080, 12600, 15120, 22320, 25200, 27000, 27720, 37800, 55440, 55800, 55860, 110880, 138600, 166320, 167580, 277200

Offset: 1

Benoit Cloitre, Aug 23 2002

nonn

Previous name was: Numbers n such that sigma(k)/k < sigma(n)/n for k=1,2,3...,n-1.

Amiram Eldar, Table of n, a(n) for n = 1..128

Cf. A004394.

lista(nn) = {vs = Set(); maxs = 0; for(n = 1, nn, news = sigma(n, -1); if (news > maxs, maxs = news; print1(n, ", "); vs = Set(concat(vs, maxs));, if (setsearch(vs, news), print1(n, ", "));););} \\ Michel Marcus, Dec 02 2013

New name and more terms from Michel Marcus, Dec 02 2013

A134672 Inverse Mobius transform of the superabundant numbers, A051731 * A004394.

Original entry on oeis.org

1, 3, 5, 9, 13, 31, 37, 57, 65, 135, 181, 277, 361, 759, 857, 1317, 1681, 2611, 5041, 10221, 15161, 25383, 27721, 55765, 110893, 166683, 277265, 333405, 554401, 666283, 720721, 1442757, 2162345, 3605283, 4324369, 7210057, 8648641, 10815843, 21621965, 36766989

Offset: 1

Gary W. Adamson, Nov 05 2007

nonn

			a(4) = 9 = (1, 1, 0, 1) dot (1, 2, 4, 6) = (1 + 2 + 0 + 6), where (1, 1, 0, 1) = row four of A051731.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A004394, A051731.

sa={}; rm=0; Do[r=DivisorSigma[1,n]/n; If[r>rm, rm=r;AppendTo[sa, n]], {n, 1, 120000}]; s={}; Do[AppendTo[s, DivisorSum[j, sa[[#]]&]], {j, 1, Length[sa]}]; s (* Amiram Eldar, Jun 28 2019 *)

Inverse Mobius transform of A004394

More terms from Amiram Eldar, Jun 28 2019

A305025 a(n) = A001221(A004394(n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11

Offset: 1

Michael De Vlieger, Jun 30 2018

Number of distinct prime factors of superabundant numbers.
Analogous to A108602 (which instead pertains to A002182, the highly composite numbers).
a(23) = 5 while A108602(23) = 4; 23 is the smallest index where this sequence differs from A108602.

			A004394(8) = 48 = 2^4*3, which has 2 distinct prime factors, so a(8)=2.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001221, A004394, A108602.

(* First, convert terms in b-file at A004394 into a list of terms: *)
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]};
s = Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ];
PrimeNu[Take[s, 105]]

A307322 Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A004394(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 3, 2, 3, 1, 1, 1, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 4, 2, 4, 1, 1, 1, 4, 1, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 4, 1, 2, 2, 4, 1, 1, 3, 4, 1, 2, 5, 1, 1, 2, 5, 1, 1, 1, 2, 5, 1, 2, 2, 5, 1, 1, 3, 5, 1, 1, 2, 2

Offset: 1

Michael De Vlieger, Apr 02 2019

Analogous to A306737.
The first 52 terms of a(n) and A306737 are identical, since the first 19 terms of A002182 and A004394 are the same, and the first two terms of row 20 are the same. a(20) = 4,2,1,1,1, while A306737(20) = 4,2,2.
Each superabundant number A004394(n) can be expressed as a product of primorials in A002110.
Row 1 = {0} by convention.
Maximum value in row n = A001221(A004394(n)).
Row n in reverse order is the conjugate of the list of the multiplicities of the prime divisors of A004394(n).

			Terms in the first rows n of this sequence, followed by the corresponding primorials whose product = A004394(n):
   n    T(n,k)       A002110(T(n,k))   A004394(n)
  -----------------------------------------------
   1:   0;              1                =     1
   2:   1;              2                =     2
   3:   1, 1;           2 * 2            =     4
   4:   2;              6                =     6
   5:   1, 2;           2 * 6            =    12
   6:   1, 1, 2;        2 * 2 * 6        =    24
   7:   2, 2;           6 * 6            =    36
   8:   1, 1, 1, 2;     2 * 2 * 2 * 6    =    48
   9:   1, 3;           2 * 30           =    60
  10:   1, 1, 3;        2 * 2 * 30       =   120
  11:   2, 3;           6 * 30           =   180
  12:   1, 1, 1, 3;     2 * 2 * 2 * 30   =   240
  13:   1, 2, 3;        2 * 6 * 30       =   360
  14:   1, 1, 2, 3;     2 * 2 * 6 * 30   =   720
  15:   1, 1, 4;        2 * 2 * 210      =   840
  ...
Row 6 = {1,1,2} since A002110(1)*A002110(1)*A002110(2) = 2*2*6 = 24 and A004394(6) = 24. The conjugate of {1,1,2} = {3,1} and 24 = 2^3 * 3^1.
Row 10 = {1,1,3} since A002110(1)*A002110(1)*A002110(3) = 2*2*30 = 120 and A004394(10) = 120. The conjugate of {1,1,3} = {3,1,1} and 120 = 2^3 * 3^1 * 5^1.

Michael De Vlieger, Table of n, a(n) for n = 1..10664 (rows 1 <= n <= 1200, flattened).
Michael De Vlieger, Relation between A307322 and A306737.

Cf. A001221, A002110, A002182, A004394, A306737.

Block[{s = Array[DivisorSigma[1, #]/# &, 10^6]}, Map[Table[LengthWhile[#, # >= i &], {i, Max@ #}] &@ If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] /. {} -> {0}] // Flatten

A340014 Numbers k in A305056 such that k * A002110(j) is in A004394 for some j >= 0.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 48, 72, 120, 144, 240, 288, 360, 720, 1440, 2160, 2880, 4320, 5040, 8640, 10080, 15120, 20160, 30240, 60480, 120960, 151200, 181440, 241920, 302400, 604800, 907200, 1209600, 1330560, 1663200, 1814400, 3326400, 6652800, 9979200, 13305600

Offset: 1

Michael De Vlieger, Dec 29 2020

nonn

Let m be a superabundant number. Since m is a product of primorials P, we may identify a greatest primorial divisor P(omega(m)) = A002110(A001221(A004394(n))).
This sequence lists the primitive quotients k = m/P(omega(m)).
Since m is a product of primorials and k is the quotient resulting from division of m by the largest primorial divisor P, this sequence is also a subset of A025487, which in turn is a subset of A055932.
We can plot all m in A004394 at (A002110(j),k), but this sequence does not accommodate all highly composite numbers; it is missing k = {36, 96, 216, 480, ...}. In contrast, k in A301414 can represent all superabundant numbers m, but a(116)=592424239959167616000 is the least k missing. Therefore in order to plot both A002182 and A004394 one must use the union of a(n) and A301414(n). One can ably plot all the terms common to both A002182 and A004394 (i.e., A166981) using k in A301414.

			Plot of (A002110(j),k) with k a term in this sequence such that A002110(j) * k is in A004394. Asterisks denote products that are in A004490.
   {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*
                           {4,48}   {5,48}    {6,48}
                            10080   110880   1441440*
                            ...     ...      ...       ...
This table is missing 7560, 83160, 1081080 at {4,36}, {5,36}, and {6,36}, respectively, which are numbers in A002182 but not in A004394. Thus, 36 is in A301414 but not in this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..2000
Michael De Vlieger, Annotated color-coded plot (x,y) = (a(n), A002110(j)) highlighting superabundant numbers.
Michael De Vlieger, Extended annotated color-coded plot showing all terms in A004394 that also appear in A224078.
Michael De Vlieger, Simple extended color-coded plot (x,y) = (a(n), A002110(m)) for coordinates between (1,0)..(225,379), with (1,0) in lower left corner.

Cf. A002110, A004394, A004490, A025487, A055932, A301414, A305056.

Block[{s = Array[DivisorSigma[1, #]/# &, 10^6], t}, t = Union@ FoldList[Max, s]; Union@ Map[#/Product[Prime@ i, {i, PrimeNu@ #}] &@ First@ FirstPosition[s, #] &, t]]

A002182 Highly composite numbers: numbers n where d(n), the number of divisors of n (A000005), increases to a record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160

Offset: 1

N. J. A. Sloane

Where record values of d(n) occur: d(n) > d(k) for all k < n.
A002183 is the RECORDS transform of A000005, i.e., lists the corresponding values d(n) for n in A002182.
Flammenkamp's page also has a copy of the missing Siano paper.
Highly composite numbers are the product of primorials, A002110. See A112779 for the number of primorial terms in the product of a highly composite number. - Jud McCranie, Jun 12 2005
Sigma and tau for highly composite numbers through the 146th entry conform to a power fit as follows: log(sigma)=A*log(tau)^B where (A,B) =~ (1.45,1.38). - Bill McEachen, May 24 2006
a(n) often corresponds to P(n,m) = number of permutations of n things taken m at a time. Specifically, if start=1, pointers 1-6, 9, 10, 13-15, 17-19, 22, 23, 28, 34, 37, 43, 52, ... An example is a(37)=665280, which is P(12,6)=12!/(12-6)!. - Bill McEachen, Feb 09 2009
Concerning the previous comment, if m=1, then P(n,m) can represent any number. So let's assume m > 1. Searching the first 1000 terms, the only indices of terms of the form P(n,m) are 4, 5, 6, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 27, 28, 31, 34, 37, 41, 43, 44, 47, 50, 52, and 54. Note that a(44) = 4324320 = P(2079,2). See A163264. - T. D. Noe, Jun 10 2009
A large number of highly composite numbers have 9 as their digit root. - Parthasarathy Nambi, Jun 07 2009
Because 9 divides all highly composite numbers greater than 1680, those numbers have digital root 9. - T. D. Noe, Jul 24 2009
See A181309 for highly composite numbers that are not highly abundant.
a(n) is also defined by the recurrence: a(1) = 1, a(n+1)/sigma(a(n+1)) < a(n) / sigma(a(n)). - Michel Lagneau, Jan 02 2012 [NOTE: This "definition" is wrong (a(20)=7560 does not satisfy this inequality) and incomplete: It does not determine a sequence uniquely, e.g., any subsequence would satisfy the same relation. The intended meaning is probably the definition of the (different) sequence A004394. - M. F. Hasler, Sep 13 2012]
Up to a(1000), the terms beyond a(5) = 12 resp. beyond a(9) = 60 are a multiples of these. Is this true for all subsequent terms? - M. F. Hasler, Sep 13 2012 [Yes: see EXAMPLE in A199337! - M. F. Hasler, Jan 03 2020]
Differs from the superabundant numbers from a(20)=7560 on, which is not in A004394. The latter is not a subsequence of A002182, as might appear from considering the displayed terms: The two sequences have only 449 terms in common, the largest of which is A002182(2567) = A004394(1023). See A166735 for superabundant numbers that are not highly composite, and A004394 for further information. - M. F. Hasler, Sep 13 2012
Subset of A067128 and of A025487. - David A. Corneth, May 16 2016, Jan 03 2020
It seems that a(n) +- 1 is often prime. For n <= 1000 there are 210 individual primes and 17 pairs of twin primes. See link to Lim's paper below. - Dmitry Kamenetsky, Mar 02 2019
There are infinitely many numbers in this sequence and a(n+1) <= 2*a(n), because it is sufficient to multiply a(n) by 2 to get a number having more divisors. (This proves Guess 0 in the Lim paper.) For n = (1, 2, 4, 5, 9, 13, 18, ...) one has equality in this bound, but asymptotically a(n+1)/a(n) goes to 1, cf. formula due to Erdős. See A068507 for the terms such that a(n)+-1 are twin primes. - M. F. Hasler, Jun 23 2019
Conjecture: For n > 7, a(n) is a Zumkeller number (A083207). Verified for n up to and including 48. If this conjecture is true, one may base on it an alternative proof of the fact that for n>7 a(n) is not a perfect square (see Fact 5, Rao/Peng arXiv link at A083207). - Ivan N. Ianakiev, Jun 29 2019
The conjecture above is true (see the proof in the "Links" section). - Ivan N. Ianakiev, Jan 31 2020
The first instance of omega(a(n)) < omega(a(n-1)) (omega = A001221: number of prime divisors) is at a(26) = 45360. Up to n = 10^4, 1759 terms have this property, but omega decreases by 2 only at indices n = 5857, 5914 and 5971. - M. F. Hasler, Jan 02 2020
Inequality (54) in Ramanujan (1915) implies that for any m there is n* such that m | a(n) for all n > n*: see A199337 for the proof. - M. F. Hasler, Jan 03 2020

			a(5) = 12 is in the sequence because A000005(12) is larger than any earlier value in A000005. - _M. F. Hasler_, Jan 03 2020

CRC Press Standard Mathematical Tables, 28th Ed, p. 61.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris 2008.
L. E. Dickson, History of Theory of Numbers, I, p. 323.
Ross Honsberger, An introduction to Ramanujan's Highly Composite Numbers, Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985
Jean-Louis Nicolas, On highly composite numbers, pp. 215-244 in Ramanujan Revisited, Editors G. E. Andrews et al., Academic Press 1988
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 88.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (obtained from A. Flammenkamp's data; first 1000 terms from T. D. Noe)
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Harold W. Ellingsen, Jr., A Fresh Look at Highly Composite Numbers, The American Mathematical Monthly, Vol. 126, No. 8 (2019), pp. 740-741.
Paul Erdős, On Highly composite numbers, J. London Math. Soc., Vol. 19 (1944), pp. 130-133, MR7,145d; Zentralblatt 61,79.
Achim Flammenkamp, Highly composite numbers.
Achim Flammenkamp, List of the first 1200 highly composite numbers.
Achim Flammenkamp, List of the first 779,674 highly composite numbers.
James Grime and Brady Haran, 5040 and other Anti-Prime Numbers, Numberphile video (2016).
Bob Hinman, Letter to N. J. A. Sloane, Aug. 1980.
Ivan N. Ianakiev, On the question "Which Highly composite numbers (A002182) are Zumkeller numbers (A083207)?".
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
Aneesh M. Koya and P. P. Deepthi, Plug and play self-configurable IoT gateway node for telemonitoring of ECG, Computers in Biology and Medicine, Vol. 112 (2019), 103359.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543; arXiv:math/0008177 [math.NT], 2000-2001.
Bill Lauritzen, Versatile Numbers -Versatile Economics.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola Magazine, Volume 54, Issue 3, (2018).
R. J. Mathar, Maple program to convert the Flammenkamp file to an OEIS b-file.
R. J. Mathar, Output of above Maple program. [Uncompresses to 9.1 MB]
Graeme McRae, Highly Composite Numbers.
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers' (Text in French).
Jean-Louis Nicolas and Guy Robin, Highly Composite Numbers by Srinivasa Ramanujan, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
Kevin O'Bryant, PlanetMath.org, Highly composite number.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915), pp. 347-409. (DOI: 10.1112/plms/s2_14.1.347, also available with an additional footnote in the PDF at http://ramanujan.sirinudi.org/Volumes/published/ram15.html)
Steven Ratering, An interesting subset of the highly composite numbers, Math. Mag., Vol. 64, No. 5 (1991), pp. 343-346.
Guy Robin, Méthodes d'optimisation pour un problème de théorie des nombres, RAIRO Informatique Théorique, Vol. 17, No. 3 (1983), pp. 239-247.
Vladimir Shevelev, On Erdős constant, arXiv:1605.08884 [math.NT], 2016.
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.
N. J. A. Sloane, Transforms.
Michel Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
Eric Weisstein's World of Mathematics, Highly Composite Number.
Wikipedia, Highly composite number.

Cf. A000005 (number of divisors), A002110, A002183, A002473, A004394, A025487, A106037, A108602, A112778, A112779, A112780, A112781, A006218, A126098, A002201, A072938, A094348, A003418, A161184, A037992 (infinitary analog), A108951, A329902, A352418.
Cf. A261100 (a left inverse).
Cf. A002808. - Peter J. Marko, Aug 16 2018
Cf. A279930 (highly composite and highly Brazilian).
Cf. A068507 (terms such that a(n)+-1 are twin primes).
Cf. A199337 (number of terms not divisible by n).

a = 0; Do[b = DivisorSigma[0, n]; If[b > a, a = b; Print[n]], {n, 1, 10^7}]
(* Convert A. Flammenkamp's 779674-term dataset; first, decompress, rename "HCN.txt": *)
a = 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[StringSplit@ #, 1] & /@ Import["HCN.txt", "Data"] (* Michael De Vlieger, May 08 2018 *)
DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,2163000}],GreaterEqual[ #1[[2]],#2[[2]]]&] [[All,1]] (* Harvey P. Dale, May 13 2022 *)
NestList[Function[last,
  Module[{d = DivisorSigma[0, last]},
   NestWhile[# + 1 &, last, DivisorSigma[0, #] <= d &]]], 1, 40] (* Steven Lu, Mar 30 2023 *)

print1(r=1); forstep(n=2,1e5,2, if(numdiv(n)>r, r=numdiv(n); print1(", "n))) \\ Charles R Greathouse IV, Jun 10 2011

v002182 = [1]/*vector for memoization*/; A002182(n, i = #v002182) ={ if(n > i, v002182 = Vec(v002182, n); my(k = v002182[i], d, s=1); until(i == n, d = numdiv(k); s<60 && k>=60 && s=60; until(numdiv(k += s) > d,); v002182[i++] = k); k, v002182[n])} \\ Antti Karttunen, Jun 06 2017; edited by M. F. Hasler, Jan 03 2020 and Jun 20 2022

is_A002182(n, a=1, b=1)={while(n>A002182(b*=2), a*=2); until(a>b, my(m=(a+b)\2, t=A002182(m)); if(tn, b=m-1, return(m)))} \\ Also used in other sequences. - M. F. Hasler, Jun 20 2022

from sympy import divisor_count
A002182_list, r = [], 0
for i in range(1,10**4):
    d = divisor_count(i)
    if d > r:
        r = d
        A002182_list.append(i) # Chai Wah Wu, Mar 23 2015

Also, for n >= 2, smallest values of p for which A006218(p)-A006318(p-1) = A002183(n). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n+1) < a(n) * (1+log(a(n))^-c) for some positive c (see Erdős). - David A. Corneth, May 16 2016
a(n) = A108951(A329902(n)). - Antti Karttunen, Jan 08 2020
a(n+1) <= 2*a(n). For cases where the equal sign holds, see A072938. - A.H.M. Smeets, Jul 10 2021
Sum_{n>=1} 1/a(n) = A352418. - Amiram Eldar, Mar 24 2022

Jun 19 1996: Changed beginning to start at 1.
Jul 10 1996: Matthew Conroy points out that these are different from the super-abundant numbers - see A004394. Last 8 terms sent by J. Lowell; checked by Jud McCranie.
Description corrected by Gerard Schildberger and N. J. A. Sloane, Apr 04 2001
Additional references from Lekraj Beedassy, Jul 24 2001

cp's OEIS Frontend

A353302 Numbers k such that A004394(k)-1 and A004394(k)+1 are twin primes.

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A332035 Indices of A004394(n) in A055932.

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A349607 a(n) is the number of divisors of the n-th superabundant number (A004394).

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A362081 Numbers k achieving record abundance (sigma(k) > 2*k) via a residue-based measure M(k) (see Comments), analogous to superabundant numbers A004394.

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A073349 Numbers m such that sigma(m)/m is equal to sigma(k)/k for some k being superabundant (A004394).

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A134672 Inverse Mobius transform of the superabundant numbers, A051731 * A004394.

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A305025 a(n) = A001221(A004394(n)).

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A307322 Irregular triangle where row n is a list of indices in A002110 with multiplicity whose product is A004394(n).

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