cp's OEIS Frontend

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

%I A240111 #37 Feb 13 2025 08:27:13
%S A240111 8,24,27,32,40,54,56,72,88,96,104,120,125,128,135,136,152,160,168,184,
%T A240111 189,200,216,224,232,243,248,250,264,270,280,296,297,312,328,343,344,
%U A240111 351,352,360,375,376,378,384,392,408,416,424,440,456,459,472,480,486
%N A240111 Numbers for which the value of the Dedekind psi function (A001615) are less than the value of the infinitary Dedekind psi function (A049417).
%C A240111 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.
%C A240111 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
%H A240111 Amiram Eldar, <a href="/A240111/b240111.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Peter J. C. Moses)
%t A240111 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 *)
%o A240111 (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
%Y A240111 Cf. A001615, A049417, A050376.
%Y A240111 Complement of A240112 within the nonsquarefree numbers (A013929).
%K A240111 nonn
%O A240111 1,1
%A A240111 _Vladimir Shevelev_, Apr 01 2014
%E A240111 More terms from _Peter J. C. Moses_, Apr 02 2014