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A002182 Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.
(Formerly M1025 N0385)

%I M1025 N0385

%S 1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,

%T 7560,10080,15120,20160,25200,27720,45360,50400,55440,83160,110880,

%U 166320,221760,277200,332640,498960,554400,665280,720720,1081080,1441440,2162160

%N Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.

%C Where record values of d(n) occur: d(n) > d(k) for all k < n.

%C A002183 is the RECORDS transform of A000005, i.e., lists the corresponding values d(n) for n in A002182.

%C Flammenkamp's page also has a copy of the missing Siano paper.

%C 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

%C 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

%C 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

%C 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

%C A large number of highly composite numbers have 9 as their digit root. - _Parthasarathy Nambi_, Jun 07 2009

%C Because 9 divides all highly composite numbers greater than 1680, those numbers have digital root 9. - _T. D. Noe_, Jul 24 2009

%C See A181309 for highly composite numbers that are not highly abundant.

%C 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]

%C 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]

%C 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

%C Subset of A067128 and of A025487. - _David A. Corneth_, May 16 2016, Jan 03 2020

%C 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

%C 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

%C 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

%C The conjecture above is true (see the proof in the "Links" section). - _Ivan N. Ianakiev_, Jan 31 2020

%C 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

%C 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

%D CRC Press Standard Mathematical Tables 28th Ed, p. 61.

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris 2008.

%D L. E. Dickson, History of Theory of Numbers, I, p. 323.

%D Ellingsen Jr, Harold W. "A Fresh Look at Highly Composite Numbers." The American Mathematical Monthly 126.8 (2019): 740-741.

%D R. Honsberger, An introduction to Ramanujan's Highly Composite Numbers, Chap. 14 pp. 193-200 Mathematical Gems III, DME no. 9 MAA 1985

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

%H Michael De Vlieger, <a href="/A002182/b002182.txt">Table of n, a(n) for n = 1..10000</a> (Obtained from A. Flammenkamp's data; first 1000 terms from _T. D. Noe_)

%H Yu. Bilu, P. Habegger, L. Kühne, <a href="https://arxiv.org/abs/1805.07167">Effective bounds for singular units</a>, arXiv:1805.07167 [math.NT], 2018.

%H Benjamin Braun, Brian Davis, <a href="https://arxiv.org/abs/1901.01417">Antichain Simplices</a>, arXiv:1901.01417 [math.CO], 2019.

%H P. Erdős, <a href="https://www.renyi.hu/~p_erdos/1944-04.pdf">On Highly composite numbers</a>, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.

%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly composite numbers</a>

%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>

%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/HCN.bz2">List of the first 779,674 highly composite numbers</a>

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=2JM2oImb9Qg">5040 and other Anti-Prime Numbers</a>, Numberphile video (2016)

%H B. Hinman, <a href="/A006480/a006480_1.pdf">Letter to N. J. A. Sloane, Aug. 1980</a>

%H Ivan N. Ianakiev, <a href="/A002182/a002182.txt">On the question "Which Highly composite numbers (A002182) are Zumkeller numbers (A083207)?"</a>

%H Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, Alexey Belyshev, <a href="https://www.emis.de/journals/JIS/VOL23/Zaikin/zaikin3.html">Enumerating Diagonal Latin Squares of Order Up to 9</a>, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.

%H Aneesh M. Koya, P. P. Deepthi, <a href="https://doi.org/10.1016/j.compbiomed.2019.103359">Plug and play self-configurable IoT gateway node for telemonitoring of ECG</a>, Computers in Biology and Medicine (2019) Vol. 112, 103359.

%H J. C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (#6, 2002), 534-543; arXiv:math/0008177 [math.NT], 2000-2001.

%H W. Lauritzen, <a href="http://www.earth360.com/math-versatile.html">Versatile Numbers -Versatile Economics</a>

%H Benny Lim, <a href="https://www.parabola.unsw.edu.au/files/articles/2010-2019/volume-54-2018/issue-3/vol54_no3_4.pdf">Prime Numbers Generated From Highly Composite Numbers</a>, Parabola Magazine, volume 54, issue 3, 2018.

%H R. J. Mathar, <a href="/A002182/a002182.mp">Maple program to convert the Flammenkamp file to an OEIS b-file</a>.

%H R. J. Mathar, <a href="/A002182/a002182.txt.gz">Output of above Maple program</a>. [Uncompresses to 9.1 MB]

%H Graeme McRae, <a href="https://web.archive.org/web/20190223125015/http://2000clicks.com/mathhelp/NumberFactorsHighlyComposite.aspx">Highly Composite Numbers</a>.

%H J.-L. Nicolas, <a href="https://doi.org/10.24033/bsmf.1676">Ordre maximal d'un element du groupe S_n de permutations et 'highly composite numbers'</a> (Text in French).

%H J.-L. Nicolas and G. Robin, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly Composite Numbers by Srinivasa Ramanujan</a>, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.

%H K. O'Bryant, PlanetMath.org, <a href="https://web.archive.org/web/20090615050926/https://planetmath.org/encyclopedia/HighlyCompositeNumber.html">Highly composite number</a>.

%H S. Ramanujan, <a href="https://doi.org/10.1112/plms/s2_14.1.347">Highly composite numbers</a>, Proceedings of the London Mathematical Society ser. 2, vol. XIV, no.1 (1915): 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)

%H S. Ratering, <a href="http://www.jstor.org/stable/2690653">An interesting subset of the highly composite numbers</a>, Math. Mag., 64 (1991), 343-346.

%H G. Robin, <a href="http://www.numdam.org/item/?id=ITA_1983__17_3_239_0">Méthodes d'optimisation pour un problème de théorie des nombres</a>, RAIRO Informatique Théorique, 17, 1983, 239-247.

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1605.08884">On Erdős constant</a>, arXiv:1605.08884 [math.NT], 2016.

%H D. B. Siano and J. D. Siano, <a href="http://wwwhomes.uni-bielefeld.de/achim/julianmanuscript3.pdf">An Algorithm for Generating Highly Composite Numbers</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H M. Waldschmidt, <a href="http://www.math.jussieu.fr/~miw/articles/pdf/LegacyRamanujan2013Text.pdf">From highly composite numbers to transcendental number theory</a>, 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly Composite Number</a>.

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Highly_composite_number">Highly composite number</a>.

%F 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

%F a(n+1) < a(n) * (1+log(a(n))^-c) for some positive c (see Erdős). - _David A. Corneth_, May 16 2016

%F a(n) = A108951(A329902(n)). - _Antti Karttunen_, Jan 08 2020

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

%t a = 0; Do[b = DivisorSigma[0, n]; If[b > a, a = b; Print[n]], {n, 1, 10^7}]

%t (* Convert A. Flammenkamp's 779674-term dataset; first, decompress, rename "HCN.txt": *)

%t 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 *)

%o (PARI) print1(r=1); forstep(n=2,1e5,2, if(numdiv(n)>r, r=numdiv(n); print1(", "n))) \\ _Charles R Greathouse IV_, Jun 10 2011

%o (PARI) v002182 = Vec(1,1000) /* Vector for memoization. */; A002182(n) = { if(n <= #v002182 && v002182[n], v002182[n], my(i=min(n, #v002182+1), k, d, s=1); if(n >= i, v002182 = Vec(v002182,n*9\8)); until(v002182[i-=1],); k = v002182[i]; while( i<n, d = numdiv(k); if(s<60 && k>=60, s=60); until(numdiv(k+=s) > d,); v002182[i+=1] = k); k)} \\ [Could be made more efficient using even larger steps.] - _Antti Karttunen_, Jun 06 2017; edited by _M. F. Hasler_, Jan 03 2020

%o (Python)

%o from sympy import divisor_count

%o A002182_list, r = [], 0

%o for i in range(1,10**4):

%o d = divisor_count(i)

%o if d > r:

%o r = d

%o A002182_list.append(i) # _Chai Wah Wu_, Mar 23 2015

%Y 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.

%Y Cf. A261100 (a left inverse).

%Y Cf. A002808. - _Peter J. Marko_, Aug 16 2018

%Y Cf. A279930 (highly composite and highly Brazilian).

%Y Cf. A068507 (terms such that a(n)+-1 are twin primes).

%Y Cf. A199337 (number of terms not divisible by n).

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E Jun 19 1996: Changed beginning to start at 1.

%E 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_.

%E Description corrected by _Gerard Schildberger_ and _N. J. A. Sloane_, Apr 04 2001

%E Additional references from _Lekraj Beedassy_, Jul 24 2001

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