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%I #9 Sep 08 2022 08:46:26
%S 1,10,6,30,132,546,270,120,840,672,1560,3960,4320,6048,9120,16380,
%T 26208,12180,36540,37380,10920,55692,34440,68040,112140,51480,63840,
%U 103320,30240,219960,273000,209160,332640,191520,1136520,393120,594720,1389960,1239840
%N a(n) is the smallest number k >= 1 with exactly n divisors d, for which sigma(k) is divisible by d*sigma(d).
%e sigma(1) = 1 = 1*sigma(1).
%e sigma(10) = 18 = 18*(1*sigma(1)) = 3*(2*sigma(2)).
%e sigma(6) = 12 = 12*(1*sigma(1)) = 2*(2*sigma(2)) = 1*(3*sigma(3)).
%e sigma(30) = 72 = 72*(1*sigma(1)) = 11*(2*sigma(2)) = 6*(3*sigma(3)) = 1*(6*sigma(6)) .
%t a[n_] := Module[{k = 1}, While[Count[Divisors[k], _?(Divisible[DivisorSigma[1, k], # * DivisorSigma[1, #]] &)] != n, k++]; k]; Array[a, 25] (* _Amiram Eldar_, May 12 2021 *)
%o (Magma) sd:=func<n,d| DivisorSigma(1,n) mod (d*DivisorSigma(1,d)) eq 0>; a:=[]; for n in [1..32] do k:=1; while #[d:d in Divisors(k)|sd(k,d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;
%o (PARI) isok(k, n) = my(sk=sigma(k)); sumdiv(k, d, (sk % (d*sigma(d))) == 0) == n;
%o a(n) = my(k=1); while (!isok(k, n), k++); k; \\ _Michel Marcus_, May 12 2021
%Y Cf. A000203, A339472, A344102.
%K nonn
%O 1,2
%A _Marius A. Burtea_, May 10 2021
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