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

%I A182862 #24 Feb 16 2025 08:33:13
%S A182862 1,2,6,12,60,360,1260,2520,27720,138600,360360,831600,10810800,
%T A182862 75675600,183783600,1286485200,24443218800,38594556000,424540116000,
%U A182862 733296564000,8066262204000,185524030692000,1693915062840000,5380196890068000,38960046445320000,166786103592108000
%N A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.
%C A182862 In other words, the sequence includes k iff A182860(k) > A182860(m) for all m < k.
%C A182862 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
%H A182862 Amiram Eldar, <a href="/A182862/b182862.txt">Table of n, a(n) for n = 1..60</a>
%H A182862 Amiram Eldar, <a href="/A182862/a182862.txt">Table of n, a(n), A182860(a(n)) for n = 1..60</a>
%H A182862 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>
%e A182862 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.
%t A182862 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 *)
%Y A182862 Subsequence of A025487, A129912, A181826, A182863. See also A034444, A085082, A182860, A182861.
%K A182862 nonn
%O A182862 1,2
%A A182862 _Matthew Vandermast_, Jan 14 2011
%E A182862 a(14)-a(26) from _Amiram Eldar_, Jan 19 2019