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.

1, 3, 5, 7, 46, 2514, 2522, 2526, 2534, 2536, 2542, 2546, 2553, 2555, 18873, 139454, 139475, 7614005, 7614010, 7614015, 7614022, 7614030, 7614033, 7614034, 7614056, 7614062, 7614066, 7614069, 7614079, 7614082, 7614086, 7614087, 7614088

Offset

1,2

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 320.

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..7613

Leroy Quet, Two number-theoretical limits (& bonus sum). Thread in NG sci.math, Oct 30 2003.

Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant

Example

a(5)=46 because d(46) - log(46) + 1 - 2*0.5772156649... = 0.016927274... is less than
abs(d(7) - log(7) + 1 - 2*0.5772156649...) = abs(-0.100341479...) with d(46)=4 and d(7)=2.

Mathematica

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 *)

Prog

(PARI)
d=1.0; n=0; \
for(j=2, 16, kmin=round(exp(j-2*Euler+1-2*d)); kmax=round(exp(j-2*Euler+1+2*d)); \
for(k=kmin, kmax, dd=abs(numdiv(k)-log(k)+1-2*Euler); \
if(dd<d, d=dd; print1(k, ", "))))
\\ Hugo Pfoertner, Dec 08 2017

Crossrefs

Cf. A000005 = number of divisors of n, A001620 = Euler's constant gamma, A089084.

Sequence in context: A146972 A102742 A363965 * A117646 A064857 A065913

Adjacent sequences: A089041 A089042 A089043 * A089045 A089046 A089047

Keyword

nonn

Author

Leroy Quet and Hugo Pfoertner, Dec 02 2003

Extensions

Terms beyond a(5) from Hans Havermann, Dec 02 2003

Status

approved