A240111 Numbers for which the value of the Dedekind psi function (A001615) are less than the value of the infinitary Dedekind psi function (A049417).

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480, 486

Offset: 1

Author

Vladimir Shevelev, Apr 01 2014

Keywords

• nonn

Comments

Numbers k for which Product_{p|k} (1 + 1/p) < Product_{q is in Q_k} (1 + 1/q), where {p} are primes, {q} are terms of A050376 and Q_k is the set of distinct q's whose product is k.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 10, 108, 1072, 10679, 106722, 1067287, 10672851, 106728514, 1067285714, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1067285... . - 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, A049417, A050376.

Complement of A240112 within the nonsquarefree numbers (A013929).

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[500], 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