A181802 -id:A181802 - cp's OEIS frontend

A181801 Number of divisors of n that are highly composite (A002182).

1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 3, 1, 3, 1

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

A divisor d of integer n is highly composite iff more multiples of (n/d) divide n than divide any smaller positive integer. This is because the number of divisors of n that are multiples of (n/d) equals the number of divisors of d, or A000005(d). (Also see example.)

a(n) = a(n+12) if n is not a multiple of 12.

			6 is a multiple of 3 highly composite integers (1, 2 and 6); therefore a(6) = 3.
As the first comment implies, there are also a(6) = 3 values of m such that 6 sets a record for number of divisors that are multiples of m. These values of m are 1, 3 and 6. All four of 6's divisors are multiples of 1; two (namely, 3 and 6) are multiples of 3; and one (namely, 6) is a multiple of 6. Each of these totals exceeds the corresponding total for any positive integer smaller than 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Eric Weisstein's World of Mathematics, Highly composite number.

Row n of A181802 gives highly composite divisors of n. Row n of A181803 gives values of m such that n sets a record for the number of its divisors that are multiples of m. Numbers that set records for a(n) are in A181806.
Cf. A002182, A181804, A181805, A322584, A352418.
Inverse Möbius transform of A322586.

v002182 = vector(128); v002182[1] = 1; \\ For memoization.
A002182(n) = { my(d,k); if(v002182[n],v002182[n], k = A002182(n-1); d = numdiv(k); while(numdiv(k) <= d, k=k+1); v002182[n] = k; k); };
A261100(n) = { my(k=1); while(A002182(k)<=n,k=k+1); (k-1); };
A322586(n) = if(1==n,1,(A261100(n)-A261100(n-1)));
A181801(n) = sumdiv(n,d,A322586(d)); \\ Antti Karttunen, Dec 20 2018

a(n) = Sum_{d|n} A322586(d). - Antti Karttunen, Dec 20 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A352418 = 2.132872... . - Amiram Eldar, Jan 01 2024

A181806 Positive integers with more highly composite divisors (A002182) than any smaller positive integer.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 120, 240, 360, 720, 5040, 10080, 15120, 30240, 60480, 151200, 166320, 332640, 665280, 1663200, 1995840, 3326400, 8648640, 17297280, 21621600, 43243200, 86486400, 129729600, 259459200, 735134400

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Numbers n such that A181801(n) > A181801(m) for all m < n. Also, numbers n such that row n of triangles A181802 and A181803 is longer than any previous row in either triangle.

Not a subsequence of A002182. The smallest positive integer which has a record number of highly composite divisors, but which is not highly composite itself, is 30240.

			12 has five divisors (namely, 1, 2, 4, 6 and 12) that are members of A002182. No positive integer smaller than 12 has more than three members of A002182 among its divisors; hence, 12 is a member of the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..153
A. Flammenkamp, List of the first 1200 highly composite numbers
Eric Weisstein's World of Mathematics, Highly composite number

A181807(n) = number of highly composite divisors of a(n) (i.e., A181801(a(n))).
Subsequence of A025487, A181804. Numbers A181804(n) such that A181805(n) increases to a record.
Includes all members of A136253.

a(20)-a(30) from Charles R Greathouse IV, Jan 14 2011

A181803 Triangle read by rows: T(n,k) is the k-th smallest divisor d of n such that n sets a record for the number of its divisors that are multiples of d.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 5, 1, 3, 6, 7, 2, 4, 8, 9, 5, 10, 11, 1, 2, 3, 6, 12, 13, 7, 14, 15, 4, 8, 16, 17, 3, 9, 18, 19, 5, 10, 20, 21, 11, 22, 23, 1, 2, 4, 6, 12, 24, 25, 13, 26, 27, 7, 14, 28, 29, 5, 15, 30, 31, 8, 16, 32, 33, 17, 34, 35, 1, 3, 6, 9, 18, 36, 37, 19, 38, 39, 10, 20, 40, 41, 7, 21, 42

Offset: 1

Matthew Vandermast, Nov 27 2010

In other words, row n contains a particular divisor d of n iff more multiples of d appear among the divisors of n than appear among the divisors of any smaller positive integer. Cf. A181808.
Row n contains A181801(n) numbers, the largest of which is n. T(n,k) * A180802(n, A181801(n)-k+1) = n.
For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, j appears in row n iff k=n/j is a member of A002182.

			First rows read: 1; 1,2; 3; 1,2,4; 5; 1,3,6; 7; 2,4,8; 9; 5,10; 11; 1,2,3,6,12;...
6 has four divisors (1, 2, 3 and 6). Of those divisors, 1, 3 and 6 appear in row 6.
a. The divisors of 6 include four multiples of 1 (1, 2, 3 and 6); two multiples of 3 (3 and 6), and one multiple of 6 (6). No positive integer smaller than 6 has more than three multiples of 1 among its divisors; hence, 1 appears in row 6. Also, no positive integer smaller than 6 has more than one multiple of 3 among its divisors, or has any multiple of 6 among its divisors. Hence, 3 and 6 both appear in row 6.
b. On the other hand, although 6 includes two multiples of 2 among its divisors (2 and 6), so does a smaller positive integer (4, whose even divisors are 2 and 4). Accordingly, 2 is not included in row 6.
The divisors of 6 that appear in row 6 are therefore 1, 3 and 6. Note that 1, 3 and 6 equal 6/6, 6/2 and 6/1 respectively, and all of the denominators in those fractions are highly composite numbers (A002182).

Eric Weisstein's World of Mathematics, Highly composite number

For the highly composite divisors of n, see A181802. See also A181808, A181809, A181810.

T(n,k) = n/(A180802(n, A181801(n)-k+1)).

A181804 List of numbers that are LCMs of some set of highly composite numbers (A002182).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120, 144, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 60480, 75600, 83160, 90720, 100800, 110880, 151200, 166320, 181440, 221760, 226800, 277200

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Numbers n such that A181801(n) is higher than A181801(d) for any proper divisor d of n. Also, numbers n such that row n of A181802 is identical to no previous row of A181802.

A002182 is a proper subsequence of this sequence. 72 is the first LCM of some set of highly composite numbers that is not itself highly composite.

			1, 2, 4, 6, 12, 24 and 36 are all highly composite numbers, and their least common multiple (LCM) is 72.  Hence, 72 is a member of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Achim Flammenkamp, List of the first 1200 highly composite numbers.
Eric Weisstein's World of Mathematics, Highly composite number.
Eric Weisstein's World of Mathematics, Least Common Multiple (LCM).
Eric Weisstein's World of Mathematics, Proper Divisor.

A181805 gives the number of highly composite divisors of a(n), or A181801(a(n)).
Subsequence of A025487.
Includes all members of A181806.
Cf. A002182, A181802.

seq[max_] := Module[{hcn = {}, hcnmax, d, dm = 0, s = {1}}, Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hcn, n]], {n, 1, max}]; hcnmax = hcn[[-1]]; Do[s = Union[Join[s, Select[LCM[hcn[[k]], s], # <= hcnmax &]]], {k, 2, Length[hcn]}]; s]; seq[300000] (* Amiram Eldar, Jun 23 2023 *)

A181805 Number of divisors of A181804(n) that are highly composite (A002182).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 6, 7, 6, 7, 8, 8, 8, 10, 11, 14, 9, 9, 12, 14, 19, 15, 20, 21, 21, 20, 15, 22, 22, 22, 21, 23, 22, 17, 23, 23, 23, 24, 25, 24, 25, 23, 23, 25, 28, 25, 27, 27, 31, 22, 27, 26, 30, 18, 29, 25, 32, 33, 28, 29, 28, 35, 25, 33, 34, 31, 31, 38, 37

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

a(n) = maximal number of members of A002182 that have a least common multiple of A181804(n). Also, a(n) = length of row A181804(n) in triangles A181802 and A181803.

4, 13 and 16 are the first three positive integers that appear nowhere in this sequence (and, therefore, nowhere in A181801). It would be interesting to know whether there are others.

			A181804(10) = 72 has exactly seven divisors that are members of A002182 (namely, 1, 2, 4, 6, 12, 24 and 36). Hence, a(10) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Achim Flammenkamp, List of the first 1200 highly composite numbers.
Eric Weisstein's World of Mathematics, Highly composite number.

A181806(m) is the m-th member of A181804 such that the value of a(n) increases to a record. See also A181807.

Cf. A002182, A181801, A181802, A181803, A181804.

seq[max_] := Module[{hcn = {}, hcnmax, d, dm = 0, s = {1}}, Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hcn, n]], {n, 1, max}]; hcnmax = hcn[[-1]]; Do[s = Union[Join[s, Select[LCM[hcn[[k]], s], # <= hcnmax &]]], {k, 2, Length[hcn]}]; Do[s[[k]] = Count[hcn, ?(Divisible[s[[k]], #] &)], {k, 1, Length[s]}]; s]; seq[300000] (* _Amiram Eldar, Jun 23 2023 *)

a(n) = A181801(A181804(n)).

More terms from Amiram Eldar, Jun 23 2023

A181807 Number of divisors of A181806(n) that are highly composite (A002182).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 14, 19, 20, 21, 22, 23, 24, 25, 28, 31, 32, 33, 35, 38, 39, 41, 48, 49, 52, 53, 57, 59, 65, 67, 69, 77, 81, 82, 86, 91, 94, 103, 105, 107, 114, 118, 122, 125, 131, 132, 135, 141, 142, 144, 145, 154, 157, 160, 163, 166, 171, 175, 180

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Also, length of row A181806(n) in triangles A181802 and A181803.

			A181806(4) = 12 has exactly five divisors (namely, 1, 2, 4, 6 and 12) that are members of A002182.  Hence, a(4) = 5.

David A. Corneth, Table of n, a(n) for n = 1..153
A. Flammenkamp, List of the first 1200 highly composite numbers
Eric Weisstein's World of Mathematics, Highly composite number

Cf. A002182, A181801, A181802, A181803, A181806.

a(n) = A181801(A181806(n)).

More terms from Amiram Eldar, Aug 29 2019 (calculated from the b-file at A181806)

cp's OEIS Frontend

A181801 Number of divisors of n that are highly composite (A002182).

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A181806 Positive integers with more highly composite divisors (A002182) than any smaller positive integer.

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A181803 Triangle read by rows: T(n,k) is the k-th smallest divisor d of n such that n sets a record for the number of its divisors that are multiples of d.

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A181804 List of numbers that are LCMs of some set of highly composite numbers (A002182).

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A181805 Number of divisors of A181804(n) that are highly composite (A002182).

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A181807 Number of divisors of A181806(n) that are highly composite (A002182).

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