A006590 a(n) = Sum_{k=1..n} ceiling(n/k). (Formerly M2522)

1, 3, 6, 9, 13, 16, 21, 24, 29, 33, 38, 41, 48, 51, 56, 61, 67, 70, 77, 80, 87, 92, 97, 100, 109, 113, 118, 123, 130, 133, 142, 145, 152, 157, 162, 167, 177, 180, 185, 190, 199, 202, 211, 214, 221, 228, 233, 236, 247, 251, 258, 263, 270, 273, 282, 287, 296, 301

Offset

1,2

Comments

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

Given the fact that ceiling(x) <= x+1, we can, using well known results for the harmonic series, easily derive that n*log(n) <= a(n) <= n*(1+log(n)) + n = n(log(n)+2). - Stefan Steinerberger, Apr 08 2006

References

Marc LeBrun, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

M. Le Brun, Email to N. J. A. Sloane, Jul 1991

Formula

a(n) = n+Sum_{k=1..n-1} tau(k). - Vladeta Jovovic, Oct 17 2002

a(n) = 1 + a(n-1) + tau(n-1), a(n) = A006218(n-1) + n. - T. D. Noe, Jan 05 2007

a(n) = a(n-1) + A000005(n) + 1 for n >= 2. a(n) = A161886(n) - A000005(n) + 1 = A161886(n-1) + 2 = A006218(n) + A049820(n) for n >= 1. - Jaroslav Krizek, Nov 14 2009

Maple

seq(add(ceil(n/j), j = 1..n), n = 1..60); # G. C. Greubel, Nov 07 2019

Mathematica

Table[Sum[Ceiling[n/i], {i, 1, n}], {n, 1, 60}] (* Stefan Steinerberger, Apr 08 2006 *)
nxt[{n_, a_}]:={n+1, a+DivisorSigma[0, n]+1}; Transpose[NestList[nxt, {1, 1}, 60]][[2]] (* Harvey P. Dale, Aug 23 2013 *)

Prog

(Haskell)
a006590 n = sum $ map f [1..n] where
   f x = y + 1 - 0 ^ r where (y, r) = divMod n x
-- Reinhard Zumkeller, Feb 18 2013
(PARI) first(n)=my(v=vector(n, i, i), s); for(i=1, n-1, v[i+1]+=s+=numdiv(i)); v \\ Charles R Greathouse IV, Feb 07 2017
(PARI) a(n) = n + sum(k=1, n-1, (n-1)\k); \\ Michel Marcus, Oct 10 2021
(Magma) [&+[Ceiling(n/j): j in [1..n]] : n in [1..60]]; // G. C. Greubel, Nov 07 2019
(Sage) [sum(ceil(n/j) for j in (1..n)) for n in (1..60)] # G. C. Greubel, Nov 07 2019
(Python)
from math import isqrt
def A006590(n): return (lambda m: n+2*sum((n-1)//k for k in range(1, m+1))-m*m)(isqrt(n-1)) # Chai Wah Wu, Oct 09 2021

Crossrefs

Cf. A000005, A006218.

Sequence in context: A060605 A325228 A278449 * A061781 A123753 A124288

Adjacent sequences: A006587 A006588 A006589 * A006591 A006592 A006593

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Stefan Steinerberger, Apr 08 2006

Status

approved