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

1,1

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A072498 A162643 A072587 * A368714 A374904 A162645

Adjacent sequences: A240109 A240110 A240111 * A240113 A240114 A240115

Keyword

nonn,changed

Author

Vladimir Shevelev, Apr 01 2014

Extensions

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

Status

approved