A108602 -id:A108602 - cp's OEIS frontend

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

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)).

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

Matthew Vandermast, Jun 08 2012

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).

			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.

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.

Peter J. Marko, Table of i, a(i) for i = 1..10022 (corresponding to first n = 584 rows of irregular triangle; using data from Flammenkamp)
A. Flammenkamp, Highly composite numbers
A. Flammenkamp, List of the first 1200 highly composite numbers
A. Flammenkamp, List of the first 779,674 highly composite numbers
Peter J. Marko, Table of n, T(n, k) by rows for n = 1..10000 (using data from Flammenkamp)
S. Ramanujan, Highly Composite Numbers

Row n has length A108602(n), n >= 2.

Cf. A000040, A002182, A124010, A212171, A318490.

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

Edited by Peter J. Marko, Aug 30 2018

A318490 Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists 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, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 7, 11, 2, 3, 5, 7

Offset: 1

Peter J. Marko, Aug 27 2018

The exponents of factors in row n are given by A212182(n).

			Triangle begins:
  0;
  2;
  2;
  2, 3;
  2, 3;
  2, 3;
  2, 3;
  2, 3;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5;
  2, 3, 5, 7;
  ...
1st row: A002182(1) = 1 so T(1,1) = 0;
2nd row: A002182(2) = 2 so T(2,1) = 2;
3rd row: A002182(3) = 4 = 2^2 so T(3,1) = 2;
4th row: A002182(4) = 6 = 2 * 3 so T(4,1) = 2 and T(4,2) = 3;
5th row: A002182(5) = 12 = 2^2 * 3 so T(5,1) = 2 and T(5,2) = 3;
6th row: A002182(6) = 24 = 2^3 * 3 so T(6,1) = 2 and T(6,2) = 3.

Peter J. Marko, Table of i, a(i) for i = 1..10022 (corresponding to first n = 584 rows of irregular triangle; using data from Flammenkamp)
A. Flammenkamp, Highly composite numbers
A. Flammenkamp, List of the first 1200 highly composite numbers
A. Flammenkamp, List of the first 779,674 highly composite numbers
Peter J. Marko, Table of n, T(n, k) by rows for n = 1..10000 (using data from Flammenkamp)
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.

Row n has length A108602(n), n >= 2.

Cf. A000040, A002182, A212182.

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

A112778 Number of prime factors (counted with multiplicity) of highly composite numbers (definition 1, A002182).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 4, 5, 5, 6, 6, 7, 6, 6, 7, 7, 8, 8, 9, 9, 10, 9, 8, 10, 10, 9, 9, 10, 10, 11, 10, 11, 11, 11, 12, 10, 10, 11, 11, 12, 11, 12, 12, 12, 13, 12, 13, 14, 13, 13, 12, 14, 12, 13, 13, 13, 14, 13, 14, 15, 14, 14, 13, 15, 15, 14, 14, 16, 14, 15, 14, 15, 16, 15, 15, 16

Offset: 1

Ray Chandler, Nov 11 2005

nonn

The values of this sequence oscillate around a slowly increasing moving average, with an amplitude roughly equal to log(a(n)): Records 1, 2, 3, ... of max(a(1..n)) - a(n) are reached at n = (9, 25, 11, 307, 1201, 7140, ...) where a(n) = (4, 8, 18, 31, 64, 169, 175, ...). - M. F. Hasler, Jan 08 2020

			A002182(8) = 48 = 2^4*3, which has 5 prime factors, counted with multiplicity, so a(8)=5.

Joerg Arndt, Table of n, a(n) for n = 1..19999
Eric Weisstein's World of Mathematics, Highly Composite Number

Cf. A002182, A002183, A108602, A112779, A112780, A112781.

A112778(n)=bigomega(A002182(n)) \\ or A112778(n)=v112778[n] (e.g., from b-file)
/* To list the records of max(a(1..n)) - a(n): */
m=r=0; for(i=1,1e4, if(mA112778(i), m=n, m-n>r, print1([i,n,r=m-n]",")))
\\ M. F. Hasler, Jan 08 2020

a(n) = A001222(A002182(n)).

A112779 Largest exponent in the prime factorization of highly composite numbers (definition 1, A002182).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 4, 2, 3, 2, 4, 3, 4, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 3, 4, 5, 4, 3, 5, 4, 6, 4, 5, 4, 5, 6, 4, 3, 5, 4, 6, 4, 5, 4, 5, 6, 4, 6, 7, 5, 4, 4, 6, 4, 5, 4, 5, 6, 4, 6, 7, 5, 4, 4, 6, 5, 5, 4, 6, 5, 6, 4, 6, 7, 5, 4, 6, 5, 7, 6, 5, 6, 4, 4, 6, 7, 5, 5, 4, 6, 6, 5, 7, 6, 5, 6, 4, 7, 6, 7, 5, 7, 6, 8

Offset: 1

Ray Chandler, Nov 11 2005

nonn

Each highly composite number can be written as the product of primorials (A002110); a(n) is also the number of primorials used in the product.

a(i) is the exponent of 2 in the prime factorization of A002182(i), cf. formula. - David A. Corneth, Aug 16 2015; edited by M. F. Hasler, Jan 03 2020

			A002182(8) = 48 = 2^4*3, which has largest exponent 4, so a(8)=4.

T. D. Noe, Table of n, a(n) for n=1..10000 (using Flammenkamp's data)
A. Flammenkamp, First 1200 highly composite numbers
Eric Weisstein's World of Mathematics, Highly Composite Number

Cf. A002110, A002182, A002183, A007814, A108602, A112778, A112780, A112781.

apply( A112779(n)=valuation(A002182(n),2), [1..99]) \\ M. F. Hasler, Jan 03 2020

a(n) = A007814(A002182(n)). - David A. Corneth, Aug 16 2015

A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 8, 2, 4, 6, 8, 12, 16, 20, 12, 14, 18, 12, 12, 12, 12, 12, 12, 12, 24, 8, 8, 8, 4, 16, 8, 16, 8, 16, 24, 16, 32, 6, 14, 30, 12, 18, 18, 24, 12, 18, 18, 24, 18, 36, 8, 14, 32, 28, 6, 24, 38, 12, 18, 36, 20, 24, 30, 40, 26, 10, 40, 20, 30, 18, 38, 26, 36, 36, 24, 24, 44, 50, 48, 14, 42

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

Among the first 10000 highly composite numbers, only in two cases a(n) < A112779(n). This happens on A002182(12) = 240 and A002182(18) = 2520. Note that A112779(n) gives the number of primorials needed when A002182(n) is expressed as a product [not as a sum] of primorials.

			For n=12, A002182(12) = 240, which is written as "11000" in primorial base (A049345) because 240 = 1*A002110(4) + 1*A002110(3) = 210+30, thus a(12) = 1+1 = 2. (Note that 240 = 30*2*2*2).
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 1+1 = 2. (Note that 2520 = 210*6*2).
For n=26, A002182(26) = 45360 = "1670000" in primorial base because 45360 = 1*A002110(6) + 6*A002110(5) + 7*A002110(4), thus a(26) = 1+6+7 = 14. (Note that 45360 = 210*6*6*6).

Cf. A002110, A002182, A276150, A324381.

Cf. also A108602, A112778, A112779, A324342.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324382(n) = A276150(A002182(n));

a(n) = A276150(A002182(n)).

a(n) >= A324381(n).

A112781 Number of highly composite numbers (definition 1, A002182) < 10^n.

Original entry on oeis.org

4, 9, 15, 20, 29, 38, 47, 56, 66, 76, 86, 95, 106, 117, 125, 135, 146, 156, 167, 177, 186, 196, 209, 219, 231, 241, 254, 267, 280, 292, 305, 316, 330, 343, 356, 368, 381, 396, 409, 423, 436, 450, 463, 476, 491, 503, 517, 530, 547, 561, 577, 593, 608, 625, 640

Offset: 1

Ray Chandler, Nov 11 2005

nonn

			a(1) = 4 since there are four highly composite numbers < 10^1 {1,2,4,6}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Highly Composite Number
A. Flammenkamp, First 1200 highly composite numbers

Cf. A002182, A002183, A108602, A112778, A112779, A112780.

Partial sums of A112780. - Lekraj Beedassy, Sep 02 2006

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

Ray Chandler, Nov 11 2005

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Highly Composite Number
A. Flammenkamp, First 1200 highly composite numbers

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

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

cp's OEIS Frontend

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

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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).

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A318490 Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).

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

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A112778 Number of prime factors (counted with multiplicity) of highly composite numbers (definition 1, A002182).

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A112779 Largest exponent in the prime factorization of highly composite numbers (definition 1, A002182).

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A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

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