

A181810


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


4



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
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OFFSET

1,2


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).


LINKS



EXAMPLE

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.


MATHEMATICA

f[hc_, n_] := Module[{k=1}, While[MemberQ[hc, n/k], k++]; k1]; 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 *)


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.


KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



