A181807 -id:A181807 - cp's OEIS frontend

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

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

A181802 Triangle read by rows: T(n,k) is k-th smallest divisor of n that is highly composite (A002182).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 24, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 36, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 24, 48

Offset: 1

Matthew Vandermast, Nov 27 2010

Row n contains A181801(n) numbers. T(n,k) * A180803(n, A181801(n)-k+1) = n.

Row n is identical to row (n+12) if n is not a multiple of 12.

			First rows read: 1; 1,2; 1; 1,2,4; 1; 1,2,6; 1; 1,2,4; 1; 1,2; 1; 1,2,4,6,12;...
8 has four divisors, of which three (1, 2 and 4) are members of A002182. Row 8 therefore reads 1, 2, 4.

Eric Weisstein's World of Mathematics, Highly composite number

See also A181804, A181805, A181806, A181807.

T(n,k) = n/(A180803(n, A181801(n)-k+1)).

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

cp's OEIS Frontend

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

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A181802 Triangle read by rows: T(n,k) is k-th smallest divisor of n that is highly composite (A002182).

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

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