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

1, 4, 8, 16, 32, 64, 128, 256, 512, 576, 1152, 1600, 2304, 4608, 3600, 6300, 7200, 18900, 20736, 32725, 14400, 28800, 50400, 56700, 108900, 57600, 100800, 111321, 176400, 129600, 226800, 229075, 360000, 630000, 435600, 333963, 518400, 1374450, 871200, 1001889

Offset: 0

Author

Marius A. Burtea, Jul 12 2024

Keywords

• nonn

Comments

Numbers of the form m = 2^(k+1), k >= 0, have exactly k divisors that are Duffinian numbers.

Examples

			Since A003624(1) = 4, a(0) = 1.
The numbers 2 and 3 have no divisors that are Duffinian numbers and 4 = A003624(1), so a(1) = 4.
		

Crossrefs

Cf. A003624, A373968.

Programs

Magma

f:=func; 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;

Mathematica

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 *)