A181803 -id:A181803 - 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

A181808 Numbers that set a record for number of even divisors: a(n) = 2*A002182(n).

Original entry on oeis.org

2, 4, 8, 12, 24, 48, 72, 96, 120, 240, 360, 480, 720, 1440, 1680, 2520, 3360, 5040, 10080, 15120, 20160, 30240, 40320, 50400, 55440, 90720, 100800, 110880, 166320, 221760, 332640, 443520, 554400, 665280, 997920, 1108800, 1330560, 1441440, 2162160, 2882880, 4324320

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

In other words, a positive integer n appears in the sequence iff more even numbers divide n than divide any positive integer smaller than 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, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182). Cf. A181803, A181809, A181810.

			a(4)=12 has exactly four even divisors (2, 4, 6 and 12).  (Note that these are precisely the numbers that are twice a divisor of A002182(4)=6; see row 6 of A027750.)  No positive integer smaller than 12 has as many as four even divisors; hence, 12 is a member of the sequence.

Eric Weisstein's World of Mathematics, Highly composite number

Numbers n such that 2 appears in row n of A181803. See also A181809, A181810.
A002183(n) gives number of even divisors of a(n).
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors.

a(n)=2*A002182(n).

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

A181809 Numbers n such that both n and n/2 are highly composite (A002182).

Original entry on oeis.org

2, 4, 12, 24, 48, 120, 240, 360, 720, 1680, 2520, 5040, 10080, 15120, 20160, 50400, 55440, 110880, 166320, 221760, 332640, 554400, 665280, 1441440, 2162160, 2882880, 4324320, 7207200, 8648640, 14414400, 17297280, 21621600, 43243200, 73513440

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

These are the numbers that set records both for total number of divisors and for number of even divisors; intersection of A002182 and A181808.

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, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182).

			The number 12 is both highly composite (A002182(5) = 12) and twice another highly composite number (12 = 2*6 = 2*A002182(4)).  It therefore has more divisors (A002183(5)=6) than any smaller positive integer, and more even divisors (A002183(4)=4) than any smaller positive integer. Since 12 is the third positive integer with the properties that define this sequence, a(3)=12.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Highly composite number

Numbers n such that 1 and 2 both appear in row n of A181803. See also A181808, A181810.
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors. Only the number 1 sets a record for its number of odd divisors and its number of total divisors.
Subsequence of A025487.

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)

A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 2, 3, 2, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 2, 2, 3, 1, 1, 6, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 1, 6, 3, 2, 1, 4, 1, 2, 1, 2, 2, 3, 1, 6, 3, 2, 4, 1, 1, 1, 1, 2, 2

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Also, largest number k such that, for each integer j from 1 to k, more multiples of j appear among the divisors of A002182(n) than appear among the divisors of any smaller positive integer.

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, n sets a record for the number of its divisors that are multiples of j iff k = n/j is highly composite (A002182).

			360 is a member of A002182, twice a member of A002182 (360/2 = 180), and three times a member of A002182 (360/3 = 120), but is not four times a member of A002182 (360/4 = 90 is not a member of A002182). Since A002182(13) = 360, a(13) = 3.
360 also sets records for the number of its divisors, the number of its divisors that are multiples of 2 (cf. A181808), and the number of its divisors that are multiples of 3, but not the number of its divisors that are multiples of 4.

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

a(n) equals the largest number k such that each number from 1 to k appears in row A002182(n) of A181803. a(n) also equals the largest number k such that each of the first k members of row A002182(n) of A056538 is highly composite.

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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A181808 Numbers that set a record for number of even divisors: a(n) = 2*A002182(n).

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

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A181809 Numbers n such that both n and n/2 are highly composite (A002182).

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

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A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

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