A240112 Numbers for which the values of the Dedekind psi function (A001615) are greater than the values of the infinitary Dedekind psi function (A049417).

4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207

Offset: 1

Author

Vladimir Shevelev, Apr 01 2014

Keywords

• nonn

Comments

The first term of A072587 that is not in this sequence is 72.

On the set of the nonsquarefree numbers (A013929) it is complement to A240111.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 29, 284, 2845, 28527, 285352, 2853422, 28534455, 285344362, 2853443344, ... . Apparently, the asymptotic density of this sequence exists and equals 0.2853443... . - Amiram Eldar, Feb 13 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Peter J. C. Moses)

Crossrefs

Cf. A001615, A013929, A049417, A072587, A240111.

Programs

Mathematica

f1[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; f2[p_, e_] := (p+1)*p^(e-1); q[1] = False; q[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct > Times @@ f1 @@@ fct]; Select[Range[250], q] (* Amiram Eldar, Feb 13 2025 *)

PARI

isok(k) = {my(f = factor(k), b); prod(i=1, #f~, (f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)) > prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));} \\ Amiram Eldar, Feb 13 2025

Extensions

More terms from Peter J. C. Moses, Apr 02 2014