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

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

Author

Matthew Vandermast, Nov 27 2010

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• 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

Crossrefs

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.

See also A181801, A181808, A181809.

Programs

Mathematica

f[hc_, n_] := Module[{k=1}, While[MemberQ[hc, n/k], k++]; k-1]; s={}; hc={}; dm = 0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hc, n]]; AppendTo[s, f[hc, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 08 2019 *)

Extensions

a(5) corrected and more terms added by Amiram Eldar, Jul 08 2019