A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

1, 2, 6, 12, 60, 360, 1260, 2520, 27720, 138600, 360360, 831600, 10810800, 75675600, 183783600, 1286485200, 24443218800, 38594556000, 424540116000, 733296564000, 8066262204000, 185524030692000, 1693915062840000, 5380196890068000, 38960046445320000, 166786103592108000

Offset

1,2

Comments

In other words, the sequence includes k iff A182860(k) > A182860(m) for all m < k.

The records for the number of distinct prime signatures are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, ... (see the link for more values). - Amiram Eldar, Jul 07 2019

Links

Amiram Eldar, Table of n, a(n) for n = 1..60

Amiram Eldar, Table of n, a(n), A182860(a(n)) for n = 1..60

Eric Weisstein's World of Mathematics, Unitary Divisor

Example

60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  This makes a total of 6 distinct prime signatures that appear among the unitary divisors of 60.  Since no positive integer smaller than 60 has more than 4 distinct prime signatures appearing among its unitary divisors, 60 belongs to this sequence.

Mathematica

f[1] = 1; f[n_] := Times @@ (Values[Counts[FactorInteger[n][[;; , 2]]]] + 1); fm = 0; s={}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Jan 19 2019 *)

Crossrefs

Subsequence of A025487, A129912, A181826, A182863. See also A034444, A085082, A182860, A182861.

Sequence in context: A254232 A335831 A189394 * A072938 A160274 A048803

Adjacent sequences: A182859 A182860 A182861 * A182863 A182864 A182865

Keyword

nonn

Author

Matthew Vandermast, Jan 14 2011

Extensions

a(14)-a(26) from Amiram Eldar, Jan 19 2019

Status

approved