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

1,1

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)

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 *)

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, A049417, A050376.

Complement of A240112 within the nonsquarefree numbers (A013929).

Sequence in context: A295661 A376142 A377844 * A301517 A374459 A381312

Adjacent sequences: A240108 A240109 A240110 * A240112 A240113 A240114

Keyword

nonn

Author

Vladimir Shevelev, Apr 01 2014

Extensions

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

Status

approved