A067128 -id:A067128 - cp's OEIS frontend

A308531 Largely composite numbers (A067128) with a unique number of divisors.

1, 4, 36, 48, 180, 720, 5040, 20160, 25200, 45360, 50400, 498960, 665280, 3603600, 6486480, 7207200, 8648640, 14414400, 32432400, 110270160, 698377680, 2095133040, 2205403200, 41902660800, 73329656400, 146659312800, 240940299600, 293318625600, 963761198400

Offset: 1

Amiram Eldar, Jun 06 2019

nonn

These are highly composite numbers (A002182) that have no other largely composite numbers with the same number of divisors.

The corresponding numbers of divisors d(a(n)) are 1, 3, 9, 10, 18, 30, 60, 84, 90, 100, ... (see the link for more values).

			4 is in the sequence since it is the only largely composite number with 3 divisors.
2 is not in the sequence since it has 2 divisors, the same as the next largely composite number 3.

Amiram Eldar, Table of n, a(n) for n = 1..114
Amiram Eldar, Table of n, a(n), d(a(n)) for n = 1..114

Cf. A000005, A002182, A067128, A308530.

s = {}; dm = 1; c = 0; nprev = 1; Do[d = DivisorSigma[0, n]; If[d == dm, c++]; If[d > dm, dm = d; If[c == 1, AppendTo[s, nprev]]; c = 1; nprev = n], {n, 1, 10^8}]; s

A002182(k) is in the sequence if A308530(k) = 1.

A308532 Highly composite numbers (A002182) with a record number of largely composite numbers (A067128) having the same number of divisors.

Original entry on oeis.org

1, 2, 6, 60, 360, 73513440, 1396755360, 4497552259200, 130429015516800, 149602080797769600, 4488062423933088000, 6133685312708553600, 184010559381256608000, 7912454053394034144000, 19709923047004539052704000, 1162885459773267804109536000, 780296143507862696557498656000

Offset: 1

Amiram Eldar, Jun 06 2019

nonn

The corresponding record numbers of divisors d(a(n)) are 1, 2, 3, 6, 9, 10, 11, 14, 16, 20, ... (see the link for more values).

			6 is in the sequence since there are a record number of 3 largely composite numbers, 6, 8, and 10 with the same number of divisors. The next record is of 60, with 6 largely composite numbers, 60, 72, 84, 90, 96, and 108, with the same number of divisors.

Amiram Eldar, Table of n, a(n) for n = 1..25
Amiram Eldar, Table of n, a(n), d(a(n)) for n = 1..25

Cf. A000005, A002182, A067128, A308530.

s = {}; dm = 1; c = 0; cm = 0; nprev = 1; Do[d = DivisorSigma[0, n]; If[d == dm, c++]; If[d > dm, dm = d; If[c > cm, cm = c; AppendTo[s, nprev]]; c = 1; nprev = n], {n, 1, 10^3}]; s

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

A034287 Numbers whose product of divisors is larger than that of any smaller number.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240

Offset: 1

Erich Friedman

It appears that 2 and 3 are the only terms in this sequence that are not in A034288. - T. D. Noe, Mar 10 2007
Is this the same as A067128?
a(n) = numbers m where record values occur in A007955(m); A007955(m) = product of divisors of m. a(n) = possible values of A174901(m) in increasing order, a(n) = the smallest numbers k such that A007955(k) = A174899(n). - Jaroslav Krizek, Apr 01 2010
Equals A067128 for the 105834 terms less than 10^150.

Amiram Eldar, Table of n, a(n) for n = 1..230 (terms below 10^10, terms 1..161 from T. D. Noe)

Cf. A007955, A007956, A034288, A034090, A034091, A067128.

divProd[n_] := Times @@ Divisors[n]; a[1] = 1; a[n_] := a[n] = Catch[For[dp = divProd[an = a[n - 1]]; an++, True, an++, If[divProd[an] > dp, Throw[an]]]]; Table[a[n], {n, 1, 52}] (* Jean-François Alcover, Feb 01 2013 *)
DeleteDuplicates[Table[{n,Times@@Divisors[n]},{n,10000}],GreaterEqual[#1[[2]],#2[[2]]]&] [[;;,1]] (* Harvey P. Dale, Oct 07 2023 *)

A007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2))
r=0;for(n=1,1e5,t=A007955(n);if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

More terms from David W. Wilson, Dec 19 2001

A034288 Product of proper divisors is larger than for any smaller number.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080

Offset: 1

Erich Friedman

Amiram Eldar, Table of n, a(n) for n = 1..228 (terms below 10^10, terms 1..100 from T. D. Noe)

Cf. A034287, A034090, A034091, A067128, A140999.

Indices of records of A007956.

maxTerm = 10^6; record = 0; Reap[For[n = 1, n <= maxTerm, n++, p = Times @@ Most[Divisors[n]]; If[p > record, record = p; Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Aug 01 2013 *)
DeleteDuplicates[Table[{n,Times@@Most[Divisors[n]]},{n,11000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Nov 21 2024 *)

A097212 Numbers n such that A076078(n) > A076078(m) for all m < n, A076078(n) being the number of sets of distinct positive integers with a least common multiple of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 36, 48, 60, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 4620, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 138600, 166320, 221760, 277200, 332640

Offset: 1

Matthew Vandermast, Aug 07 2004

nonn

RECORDS transform of A076078. All highly composite numbers (A002182) are members. All members belong to A025487 and A067128.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A140999 (intersection of A025487 and A067128). - Matthew Vandermast, Oct 11 2008

f[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu[n/d](2^DivisorSigma[0, d] - 1))]; b = 0; l = {}; Do[c = a[n]; If[c > b, b = c; AppendTo[l, n]], {n, 10^6}]; l (* Robert G. Wilson v, Aug 13 2004 *)

More terms from Robert G. Wilson v, Aug 13 2004

A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

Original entry on oeis.org

3, 6, 12, 18, 24, 36, 48, 72, 96, 108

Offset: 1

Vladimir Shevelev, May 13 2016

Theorem. Ramanujan's largely composite numbers (A067128) having the greatest prime divisor p_k = prime(k) do not exceed Product_{2 <= p <= p_k} p^((2*ceiling(log_p(p_(k + 1)) - 1).
Proof. Let N be in A067128 with prime power factorization 2^l_1 * 3^l_2 * ... * p_k^l_k.
First let us show that l_1 <= 2x_1-1 such that 2^x_1 > p_(k+1).
Indeed, consider N_1 = 2^(l_1-x_1)*3^l_2*...*p_k^l_k*p_(k+1).
Since 2^x_1 > p_(k+1) then N_1But d(N_1) > d(N) if l_1 >= 2*x_1, so l_1 <= 2x_1-1.
Analogously we find l_i <= 2x_i-1 if p_i^x_i > p_(k+1), i <= k.
Therefore N <= 2^(2*x_1-1)*3^(2*x_2-1)*...* p_k^(2*x_k-1) and the theorem easily follows.
QED
The inequality of the theorem gives a way to find the full sequence for every p_k. In particular, in case p_k = 2 we have the sequence {2, 4, 8}. For other cases see A273016, A273018.

Cf. A067128, A065119 (the intersection of these two sequences is the present sequence). Cf. also A003586, A273016, A273018.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Select[a, FactorInteger[#][[-1, 1]] == 3 &] (* Michael De Vlieger, May 13 2016 *)

A273016 Ramanujan's largely composite numbers having 5 as the greatest prime divisor.

Original entry on oeis.org

10, 20, 30, 60, 90, 120, 180, 240, 360, 480, 540, 600, 720, 1080, 1440, 2160, 4320

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 13 2016

By the theorem in comment in A273015, the sequence is finite.

Cf. A067128, A273015.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Select[a, FactorInteger[#][[-1, 1]] == 5 &] (* Michael De Vlieger, May 13 2016 *)

A116998 Numbers having no fewer distinct prime factors than any predecessor; a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 210, 330, 390, 420, 462, 510, 546, 570, 630, 660

Offset: 1

Reinhard Zumkeller, Apr 03 2006

A001221(a(n)) <= A001221(a(n+1));
A002110 is a subsequence.
The unitary version of Ramanujan's largely composite numbers (A067128), numbers having no fewer unitary divisors than any predecessor. - Amiram Eldar, Jun 08 2019
Called omega-largely composite numbers by Erdős and Nicolas (1981). - Amiram Eldar, Jun 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Alois P. Heinz)
Paul Erdős and Jean-Louis Nicolas, Sur la fonction: nombre de facteurs premiers de n, L'Enseignement Math., Vol. 27 (1981), pp. 3-27; alternative link.

Cf. A029744, A067128.

Cf. A001221 (omega), A002110 (primorial numbers).

a:= proc(n) option remember; local k, t;
      t:= nops(ifactors(a(n-1))[2]);
      for k from 1+a(n-1) while nops(ifactors(k)[2])Alois P. Heinz, Oct 05 2012

a[1] = 1; a[n_] := a[n] = For[nu = PrimeNu[a[n-1]]; k = a[n-1]+1, True, k++, If[PrimeNu[k] >= nu, Return[k]]]; Array[a, 80] (* Jean-François Alcover, Apr 11 2017 *)

A273018 Ramanujan's largely composite numbers having 7 as the greatest prime divisor.

Original entry on oeis.org

84, 168, 336, 420, 504, 630, 672, 840, 1260, 1680, 2520, 3360, 3780, 4200, 5040, 7560, 10080, 12600, 15120, 20160, 25200, 30240, 37800, 40320, 45360, 50400, 75600, 151200

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 13 2016

By the theorem in A273015, the sequence is finite.

Cf. A067128, A273015, A273016.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Select[a, FactorInteger[#][[-1, 1]] == 7 &] (* Michael De Vlieger, May 13 2016 *)

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A308531 Largely composite numbers (A067128) with a unique number of divisors.

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A308532 Highly composite numbers (A002182) with a record number of largely composite numbers (A067128) having the same number of divisors.

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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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PARI

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PARI

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A034287 Numbers whose product of divisors is larger than that of any smaller number.

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A034288 Product of proper divisors is larger than for any smaller number.

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A097212 Numbers n such that A076078(n) > A076078(m) for all m < n, A076078(n) being the number of sets of distinct positive integers with a least common multiple of n.

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A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

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A273016 Ramanujan's largely composite numbers having 5 as the greatest prime divisor.

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