A085831 -id:A085831 - cp's OEIS frontend

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

Robert G. Wilson v, Sep 21 2000

nonn

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

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.

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

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

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

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

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

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

A153876 a(n) = Sum_{i=2^(n-1)..2^n-1} sigma_0(i), sigma_0(i) number of divisors of n, n positive integer.

Original entry on oeis.org

1, 4, 11, 29, 68, 160, 364, 820, 1813, 3981, 8674, 18782, 40387, 86443, 184232, 391188, 827787, 1746443, 3674573, 7712561, 16151933, 33757505, 70422235, 146659055, 304947023, 633152157, 1312820598, 2718674046, 5623413203, 11618957217, 23982175093, 49452872529

Offset: 1

Ctibor O. Zizka, Jan 03 2009

This sequence tells how many binary numbers with n digits are there in the multiplication matrix [1,...,2^n -1]x[1,...,2^n -1]. In general, counting how many base-B numbers of length n are there in the multiplication matrix [1,...,B^n -1]x[1,...,B^n -1] gives a(n)= sum_{i=B^(n-1),(B^n)-1} sigma_0(i). Besides this motivation it is interesting to see the behavior of partial sums of sigma_0(i) on growing intervals : a(n)= sum_{i=f(n-1),f(n)} sigma_0(i).

Chai Wah Wu, Table of n, a(n) for n = 1..76

Cf. A000005, A006218, A095256, A140480, A346730.

a(n) = sum(i=2^(n-1), 2^n-1, numdiv(i)); \\ Michel Marcus, Oct 10 2021

from math import isqrt
def A153876(n): return ((t:=isqrt(b:=(1<Chai Wah Wu, Oct 23 2023

a(n) = A085831(n) - A085831(n-1)-1. - R. J. Mathar, Jan 05 2009

a(n) = Sum_{k>=1} k * A346730(n,k). - Alois P. Heinz, Aug 01 2021

a(14)-a(32) from Alois P. Heinz, Aug 01 2021

A062550 a(n) = Sum_{k = 1..2n} floor(2n/k).

Original entry on oeis.org

0, 3, 8, 14, 20, 27, 35, 41, 50, 58, 66, 74, 84, 91, 101, 111, 119, 127, 140, 146, 158, 168, 176, 186, 198, 207, 217, 227, 239, 247, 261, 267, 280, 292, 300, 312, 326, 332, 344, 356, 368, 377, 391, 399, 411, 425, 435, 443, 459, 467, 482, 492, 502, 514, 528

Offset: 0

Henry Bottomley, Jun 26 2001

nonn

The sequence A006218 : Sum_{i=1..n} floor(n/i) = Sum_{i=1..n} sigma_0(i). Sigma_0(i) is A000005. Sequences of the type : Sum_{i=1..f(n)} floor(f(n)/i)= Sum_{i=1..f(n)} sigma_0(i). This sequence a(n)= A006218(2*n). [Ctibor O. Zizka, Mar 21 2009]

For n > 0: row sums of the triangle in A013942. - Reinhard Zumkeller, Jun 04 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A013942, A156745, A085831, A153816, A153817, A153876.

a062550 0 = 0
a062550 n = sum $ a013942_row n  -- Reinhard Zumkeller, Jun 04 2013

Table[Total[Floor[2*n/Range[2*n]]], {n, 0, 100}] (* T. D. Noe, Jun 12 2013 *)

a(n) = sum(k=1, 2*n, (2*n)\k); \\ Michel Marcus, Oct 09 2021

from math import isqrt
def A062550(n): return (lambda m: 2*sum(2*n//k for k in range(1, m+1))-m*m)(isqrt(2*n)) # Chai Wah Wu, Oct 09 2021

a(n) = A006218(2n) = A056549(n)+A006218(n) = a(n-1)+A000005(2n-1)+A000005(2n)

Data corrected for n > 30 by Reinhard Zumkeller, Jun 04 2013

A175346 a(n) = Sum_{k=1..n^2} d(k), d(k) = number of divisors of k (A000005).

Original entry on oeis.org

1, 8, 23, 50, 87, 140, 201, 280, 373, 482, 605, 746, 897, 1070, 1261, 1466, 1689, 1932, 2189, 2468, 2761, 3074, 3405, 3764, 4127, 4518, 4925, 5360, 5807, 6276, 6757, 7262, 7789, 8342, 8915, 9502, 10107

Offset: 1

Ctibor O. Zizka, Apr 17 2010

nonn

Generalized sequence: Sum_{k=1..T(n)} d(k). In this sequence T(n)=n^2, in A085831 T(n)=2^n, in A006218 T(n)=n. Other examples not in the OEIS: T(n)=p(n) n-th prime, T(n)=n*(n+1)/2 n-th triangular number, T(n)= F(n) n-th Fibonacci number, etc.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A085831, A000005, A006218.

Table[Sum[DivisorSigma[0, k], {k, 1, n^2}], {n, 1, 80}] (* Carl Najafi, Aug 21 2011 *)

a(n)=sum(k=1,n^2,numdiv(k)) \\ Charles R Greathouse IV, Aug 21 2011

def A175346(n): return (m:=n**2)+(sum(m//k for k in range(2,n+1))<<1) # Chai Wah Wu, Oct 24 2023

a(n) ~ 2n^2 log n. [Charles R Greathouse IV, Aug 21 2011]

a(n) = n^2 + 2*Sum_{k=2..n} floor(n^2/k). - Chai Wah Wu, Oct 24 2023

More terms from Carl Najafi, Aug 21 2011

A158568 a(n) = Sum_{i=1..Fibonacci(n)} sigma_0(i) where sigma_0(n) is A000005(n).

Original entry on oeis.org

1, 1, 3, 5, 10, 20, 37, 70, 127, 231, 413, 746, 1307, 2295, 4010, 6957, 12031, 20712, 35514, 60718, 103500, 175989, 298539, 505399, 853777, 1439856, 2424299, 4075479, 6841787, 11470592, 19207624, 32126763, 53678285

Offset: 1

Ctibor O. Zizka, Mar 21 2009

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..106

Cf. A000005, A000045, A006218, A085831, A062550, A153876, A153817, A153816.

with(combinat):with(numtheory): A158568 := proc(n) return add(tau(i),i=1..fibonacci(n)): end: seq(A158568(n),n=1..20); # Nathaniel Johnston, May 09 2011

Module[{nn=33,f,d},f=Fibonacci[nn];d=DivisorSigma[0,Range[f]];Table[ Total[ Take[d,n]],{n,Fibonacci[Range[nn]]}]] (* Harvey P. Dale, Apr 29 2018 *)

a(n) = sum(k=1, fibonacci(n), numdiv(k)); \\ Michel Marcus, Feb 12 2019

from math import isqrt
def A153568(n):
    a, b, = 0, 1
    for _ in range(n): a, b = b, a+b
    return (lambda m: 2*sum(a//k for k in range(1, m+1))-m*m)(isqrt(a)) # Chai Wah Wu, Oct 09 2021

a(16)-a(33) from Nathaniel Johnston, May 09 2011

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A153876 a(n) = Sum_{i=2^(n-1)..2^n-1} sigma_0(i), sigma_0(i) number of divisors of n, n positive integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A062550 a(n) = Sum_{k = 1..2n} floor(2n/k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A175346 a(n) = Sum_{k=1..n^2} d(k), d(k) = number of divisors of k (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A158568 a(n) = Sum_{i=1..Fibonacci(n)} sigma_0(i) where sigma_0(n) is A000005(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions