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A373969 The smallest number k whose divisors include exactly n Duffinian numbers (A003624).

%I A373969 #10 Aug 06 2024 05:43:59
%S A373969 1,4,8,16,32,64,128,256,512,576,1152,1600,2304,4608,3600,6300,7200,
%T A373969 18900,20736,32725,14400,28800,50400,56700,108900,57600,100800,111321,
%U A373969 176400,129600,226800,229075,360000,630000,435600,333963,518400,1374450,871200,1001889
%N A373969 The smallest number k whose divisors include exactly n Duffinian numbers (A003624).
%C A373969 Numbers of the form m = 2^(k+1), k >= 0, have exactly k divisors that are Duffinian numbers.
%e A373969 Since A003624(1) = 4, a(0) = 1.
%e A373969 The numbers 2 and 3 have no divisors that are Duffinian numbers and 4 = A003624(1), so a(1) = 4.
%t A373969 f[n_] := DivisorSum[n, 1 &, CompositeQ[#] && CoprimeQ[#, DivisorSigma[1, #]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[30, 10^7] (* _Amiram Eldar_, Jul 19 2024 *)
%o A373969 (Magma) f:=func<n|n ne 1 and not IsPrime(n) and Gcd(n,DivisorSigma(1,n)) eq 1>; a:=[]; for n in [0..38] 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 A373969 Cf. A003624, A373968.
%K A373969 nonn
%O A373969 0,2
%A A373969 _Marius A. Burtea_, Jul 12 2024