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a(n) is the smallest number k with exactly n of its divisors in A037197.
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%I #9 Jun 25 2023 18:26:36

%S 1,2,8,24,96,312,600,2100,1800,5400,4200,8400,12600,25200,16800,58800,

%T 50400,67200,93600,131040,201600,252000,218400,468000,882000,873600,

%U 1386000,1528800,2646000,655200,2217600,2772000,4233600,2620800,9374400,1965600,3276000

%N a(n) is the smallest number k with exactly n of its divisors in A037197.

%e 1 has only the divisor 1 = A037197(1), so a(1) = 1.

%e 2 has divisors 1 = A037197(1) and 2 = A037197(2), so a(2) = 2.

%e 3, 4, 5, 6, 7 do not have three divisors in A037197 and 8 has divisors 1 = A037197(1), 2 = A037197(2), 8 = A037197(3), so a(3) = 8.

%t seq[len_, kmax_] := Module[{s = Table[0, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = DivisorSum[k, 1 &, DivisorSigma[0, DivisorSigma[1, #]] == DivisorSigma[0, #] &]; If[ind <= len && s[[ind]] == 0, c++; s[[ind]] = k]; k++]; s]; seq[20, 10^6] (* _Amiram Eldar_, May 05 2023 *)

%o (Magma) f:=func<n|#Divisors(DivisorSigma(1,n)) eq #Divisors(n)>; a:=[]; for n in [1..37] do k:=1; while #[d:d in Divisors(k)|f(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

%Y Cf. A000005, A000203, A037197, A062068.

%K nonn

%O 1,2

%A _Marius A. Burtea_, May 03 2023