0,2

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

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.

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 *)

(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;

Cf. A003624, A373968.

Sequence in context: A307870 A330873 A233442 * A046055 A186949 A020707

Adjacent sequences: A373966 A373967 A373968 * A373970 A373971 A373972

Marius A. Burtea, Jul 12 2024

approved