A111490 a(n) = n + Sum_{k=1..n} (n mod k). Row sums of A372727.

0, 1, 2, 4, 5, 9, 9, 15, 16, 21, 23, 33, 29, 41, 45, 51, 52, 68, 65, 83, 81, 91, 99, 121, 109, 128, 138, 152, 152, 180, 168, 198, 199, 217, 231, 253, 234, 270, 286, 308, 298, 338, 326, 368, 372, 384, 404, 450, 422, 463, 470, 500, 506, 558, 546, 584, 576, 610, 636

Offset: 0

Author

Paolo P. Lava and Giorgio Balzarotti, Nov 21 2005

Keywords

• nonn
• easy

Comments

If the binary operation mod is defined n mod k = n if k = 0 and otherwise n - k*floor(n/k), as recommended in 'Concrete Mathematics' by Graham et. al. (p. 82), then a(n) = Sum_{k=0..n} (n mod k), for n >= 0. This definition is for example implemented in Sage, but not in Python. - Peter Luschny, Jul 19 2024

Previous name was "Sum of the element of the antidiagonals of the numerical array M(m, n) defined as follows. First row (M11, M12, ..., M1n): 1, 1, 1, 1, 1, 1, ... (all 1's). Second row (M21, M22, ..., M2n): 1, 2, 1, 2, 1, 2, ... (sequence 1, 2 repeated). Third row (M31, M32, ..., M3n): 1, 2, 3, 1, 2, 3, 1, 2, 3, ... (sequence 1, 2, 3 repeated). Fourth row (M41, M42, ..., M4n): 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, ... (sequence 1, 2, 3, 4 repeated). And so on."

Then the sequence is M(1,1), M(1,2) + M(2,1), M(1,3) + M(2,2) + M(3,1), etc., a(n) = Sum_{i=1..n} M(i, n-i+1).

This means: a(n) are the antidiagonal sums of the numerical array defined by M(n, k) = 1 + (k-1) mod n. - Michel Marcus, Sep 23 2013

The successive determinants of the arrays are the factorial numbers (A000142). - Robert G. Wilson v

Examples

			If the mod operation is defined according CMath, and n = 11, then the list
[n mod k : k = 0..n] = [11, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0], and the total of this list is a(11) = 33.
		

References

• Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, 34th printing 2022.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Partial sums of A033879. - Gionata Neri, Sep 10 2015

Cf. A000142, A004125, A372727 (triangle).

Programs

Maple

A111490:=n->add(n mod i, i=1..n+1): seq(A111490(n), n=0..100); # Wesley Ivan Hurt, Dec 05 2014
seq(n + add(irem(n, k), k = 2..n-1), n = 0..58); # Peter Luschny, Jul 19 2024

Mathematica

t = Table[Flatten@Table[Range@n, {m, Ceiling[99/n]}], {n, 99}]; f[n_] := Sum[ t[[i, n - i + 1]], {i, n}]; Array[f, 58] (* Robert G. Wilson v, Nov 22 2005 *)
(* to view table *) Table[Flatten@Table[Range@n, {m, Ceiling[40/n]}], {n, 10}] // TableForm

PARI

vector(100, n, n + sum(k=2, n, n % k)) \\ Altug Alkan, Oct 12 2015

PARI

a(n) = sum(k=1, n, 2*k-sigma(k)); \\ Michel Marcus, Oct 11 2015

Python

from math import isqrt
def A111490(n): return n*(n+1)+((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Nov 01 2023

Python

def a(n): return sum(n % k if k else n for k in range(n))
print([a(n) for n in range(59)])  # Peter Luschny, Jul 19 2024

SageMath

def a(n): return sum(n.mod(k) for k in range(n))
print([a(n) for n in srange(59)])  # Peter Luschny, Jul 19 2024

Formula

a(n) = n + A004125(n). - Juri-Stepan Gerasimov, Aug 30 2009

a(n) = Sum_{i=1..n+1} (n mod i). - Wesley Ivan Hurt, Dec 05 2014

G.f.: 2*x/(1-x)^3 - (1-x)^(-1)*Sum_{k>=1} k*x^k/(1-x^k). - Robert Israel, Oct 11 2015

a(n) = (1 - Pi^2/12) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 04 2023

Extensions

Edited and extended by Robert G. Wilson v, Nov 22 2005

Prepending a(0) = 0 and new name using a formula of Juri-Stepan Gerasimov by Peter Luschny, Jul 19 2024