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A089044 Numbers n such that abs(d(n) - log(n) + 1 - 2*gamma) is a decreasing sequence, where d(n) is the number of divisors A000005(n) and gamma is Euler's constant A001620.

%I A089044 #30 Feb 16 2025 08:32:51
%S A089044 1,3,5,7,46,2514,2522,2526,2534,2536,2542,2546,2553,2555,18873,139454,
%T A089044 139475,7614005,7614010,7614015,7614022,7614030,7614033,7614034,
%U A089044 7614056,7614062,7614066,7614069,7614079,7614082,7614086,7614087,7614088
%N A089044 Numbers n such that abs(d(n) - log(n) + 1 - 2*gamma) is a decreasing sequence, where d(n) is the number of divisors A000005(n) and gamma is Euler's constant A001620.
%D A089044 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 320.
%H A089044 Hugo Pfoertner, <a href="/A089044/b089044.txt">Table of n, a(n) for n = 1..7613</a>
%H A089044 Leroy Quet, <a href="http://groups.google.com/group/sci.math/browse_thread/thread/a04e40c70f0f220b/cf00a625948fa1d3">Two number-theoretical limits (& bonus sum).</a> Thread in NG sci.math, Oct 30 2003.
%H A089044 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>
%e A089044 a(5)=46 because d(46) - log(46) + 1 - 2*0.5772156649... = 0.016927274... is less than
%e A089044 abs(d(7) - log(7) + 1 - 2*0.5772156649...) = abs(-0.100341479...) with d(46)=4 and d(7)=2.
%t A089044 f[n_] := N[ Abs[ DivisorSigma[0, n] - Log@ n + 1 - 2 EulerGamma], 32]; k = 1; lst = {}; mx = Infinity; While[k < 8000000, a = f@k; If[a < mx, mx = a; AppendTo[lst, k]]; k++]; lst (* _Robert G. Wilson v_, Dec 11 2017 *)
%o A089044 (PARI)
%o A089044 d=1.0;n=0;\
%o A089044 for(j=2,16,kmin=round(exp(j-2*Euler+1-2*d));kmax=round(exp(j-2*Euler+1+2*d));\
%o A089044 for(k=kmin,kmax,dd=abs(numdiv(k)-log(k)+1-2*Euler);\
%o A089044 if(dd<d,d=dd;print1(k,", "))))
%o A089044 \\ _Hugo Pfoertner_, Dec 08 2017
%Y A089044 Cf. A000005 = number of divisors of n, A001620 = Euler's constant gamma, A089084.
%K A089044 nonn
%O A089044 1,2
%A A089044 _Leroy Quet_ and _Hugo Pfoertner_, Dec 02 2003
%E A089044 Terms beyond a(5) from _Hans Havermann_, Dec 02 2003