A051046 Numbers k for which pi(k) > k/(H_k - 3/2), where pi is the prime-counting function and H_k is the k-th harmonic number.

1, 109, 113, 114, 199, 200, 201, 241, 242, 271, 277, 281, 282, 283, 284, 285, 286, 287, 288, 289, 293, 294, 295, 313, 317, 318, 319, 443, 444, 445, 449, 450, 451, 457, 458, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476

Offset

1,2

Comments

Panaitopol (1999) proved that all the numbers >= 1429 are in this sequence. - Amiram Eldar, Apr 12 2022

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..2500

Laurenţiu Panaitopol, Several Approximations of pi(x), Math. Ineq. Appl., Vol. 2, No. 3 (1999), pp. 317-324.

Eric Weisstein's World of Mathematics, Harmonic Number.

Eric Weisstein's World of Mathematics, Prime Counting Function.

Mathematica

ppQ[n_]:=PrimePi[n]-n/(HarmonicNumber[n]-3/2)>0; Select[Range[500], ppQ] (* Harvey P. Dale, Feb 16 2012 *)

Prog

(Haskell)
a051046 n = a051046_list !! (n-1)
a051046_list = filter
   (\x -> fromIntegral (a000720 x) > hs !! (x - 1)) [1..]
   where hs = zipWith (/)
              [1..] $ map (subtract 1.5) $ scanl1 (+) $ map (1 /) [1..]
-- Reinhard Zumkeller, Mar 20 2014

Crossrefs

Cf. A000720, A001008, A002805.

Sequence in context: A253438 A263194 A231701 * A196667 A196673 A159027

Adjacent sequences: A051043 A051044 A051045 * A051047 A051048 A051049

Keyword

nonn

Author

Eric W. Weisstein

Status

approved