A057494 a(n) = Sum_{k = 1..10^n} d(k) where d(n) = number of divisors of n (A000005).

1, 27, 482, 7069, 93668, 1166750, 13970034, 162725364, 1857511568, 20877697634, 231802823220, 2548286736297, 27785452449086, 300880375389757, 3239062263181054, 34693207724724246, 369957928177109416, 3929837791070240368, 41600963003695964400, 439035480966899467508

Offset: 0

Author

Robert G. Wilson v, Sep 21 2000

Keywords

• nonn

Comments

The Polymath project describes an algorithm for computing a(n) in time O(2.154...^n), see Tao, Croot, and Helfgott link. - Charles R Greathouse IV, Apr 16 2012

Links

• Henri Lifchitz, Table of n, a(n) for n = 0..36
• Terence Tao, Ernest Croot III, and Harald Helfgott, Deterministic methods to find primes, Mathematics of Computation, 81 (2012), 1233-1246. arXiv:1009.3956, [math.NT], 2010-2012.

Crossrefs

Cf. A006218, A050226, A085567, A085829, A085831.

Programs

Mathematica

k = s = 0; Do[ While[ k < 10^n, k++; s = s + DivisorSigma[ 0, k ] ]; Print[s], {n, 0, 8} ]

PARI

a(n) = sum(k=1, 10^n, numdiv(k)); \\ Michel Marcus, Feb 19 2017

Python

from math import isqrt
def A057494(n): return -(s:=isqrt(m:=10**n))**2+(sum(m//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023

Formula

a(n) = A006218(10^n). - Max Alekseyev, May 10 2009

Extensions

a(10)-a(16) from Max Alekseyev, Jan 25 2010

a(17)-a(19) from Donovan Johnson, Dec 26 2012

a(20)-a(27) from Hiroaki Yamanouchi, Sep 22 2015

a(28)-a(36) from Henri Lifchitz, Feb 19 2017